Neoverse

I thought I would share what I did to earn this achievement.
I used Helena to do this mainly using the White Lion guardian.

I used these gears. Oni Mask, Dark Energy, and Jack O'Lantern for the 30 skill points.
I used this method to get all the way through Transcended lv 15.

Use either the Offensive or Defensive type starting decks to get the Guardian Selection card. I am not a fan of the "Your Name Is" card from the Hybrid deck for selecting the guardian. Unless of course the Daily deck is quite good.

There are a few skills I highly suggest going for first to help out. The skills Establish Command and Beastify really help since they will give you armor for the things you'll end up doing anyways with Helena. Also, there are other skills that give armor and the skills that give you extra mana at the beginning of turn/battle are quite worth going for if they are there. It is randomized what shows up.

Cards to look for that will really help from the battle rewards are anything that will activate your guardians ability (especially the ones that activate your guardian multiple times like Beastify and Multple Orders) or you gain armor when the guardian activates like the Dragon Scale.
Constants like Master of the Guardians will help since it gives you a free guardian activation at the beginning of your turn. Or Bless of Earth where you get more Mana at the beginning of the turn. Or even like Getting Used To Loneliness to gain armor whenever you use mana.

I was able to activate the guardians ability 4 or more times plus gain 20 to 30 armor per round . This helps mow through most battles quickly and reliably. The only battles that give trouble are those with a monster that can dismiss all of your armor. Still get as much armor each turn you can because the monsters that attack usually attack first before it gets to the one that cancels your armor.

I didn't go for the loop deck because trying to get either the cards needed are getting rid of all of the extra cards took too much time/luck.

Transcended Universe Levels 1-10 are available to select immediately after you have 'beaten' the game for the first time. They are activated before you begin the run, each one adding a debuff to you or buffs to your enemies to make the game more of a challenge. Levels 11-15 are obtained by beating the level below, so there will be one run required per level here, which will help you with your fight 1000 battles achievement. They are also tracked per character, so beating level 10 as Naya doesn't unlock level 11 for Claire and Helena as well.

That being said, I would recommend that you choose Helena as your character for this, as she is by far the best of the 3 when it comes to survivability, and is no slouch on the damage front either. I would recommend that you use the two pieces of gear in your middle and bottom slots as below. Do not take the cube, it is not necessary, and why fight three diffcult bosses when you don't have to?

AI White Candy (Bottom) - This gives you access to the Lab, where you can heal between battles if necessary, remove and evolve cards if you have the spare Gold.

Backpack (Middle) - This gives you +1 Mana per turn, which cancels out the debuff from the level of difficulty.

For the top slot, I would recommend either the Oni Mask, which will give you an extra 10 skill points which can help to level the playing field a little, or the Scouter, which will help you be able to use the Data Store and Lab more frequently than you would otherwise. If you choose to take the Scouter, make sure that you are making the Precision kills, because you will want that Gold. Below is the effect of each level from 1-15, with comments as relevant.

Level 1 - Enemies Deal 20% more damage.
Level 2 - From Chapter 2 onwards, card rewards can be Hindrance type cards.
Level 3 - No more free retry. If you mess up, it will cost Trophies to try again.
Level 4 - Enemies start with 5 Armour.
Level 5 - Your Starting Deck will have the Hindrance Card Malware added. This isn't that bad, it just blocks a slot in your hand for a turn.
Level 6 - You now only can hold 1 item. (though this can be increased by the Backpack skill to 2 or 3 if it is Eviolved, I would recommend doing this if you can).
Level 7 - Start with 10% less Max HP
Level 8 - Enemies have 20% more HP
Level 9 - Receive 1 Damage at the end of every one of your turns. This is applied first, so take this into account if trying to perform a Parry.
Level 10 - No more HP Recovery after Boss Battles. This is a problem, but it can be mitigated with the medkit in the Lab, or by taking the skill that recovers 30% HP on entry to a Boss Battle.
Level 11 - Skill Points in the store are 30% more expensive (hence why the Oni Mask can be a good choice).
Level 12 - Mana reduced by 1 on the first turn of battle (cancelled out by the Backpack gear if taken).
Level 13 - Combos will always be 5 cards long. This means fewer double damage attacks, and more luck needed to hit each one. Annoying, but not awful, as your Guardian will be the main source of damage.
Level 14 - Remaining Armour is halved at the end of each (Enemy) turn. This is easily the worst one, as the strategy for stages 1-13 doesn't work anymore.
Level 15 - Lose 5 from Max HP for every Boss Battle finished. This isn't too bad, as you want to not be talking HP damage if you can avoid it, but takes away some of the slack that you had available, down to about 1 or 2 attacks by the end.

If you're going to try the loop deck as detailed in the solution for beating Challenge Mode, or Helena's transcendence, you can stop reading here and try that. I don't know how well it might work, but I used the stratgey below, which doesn't use a loop deck.

For Levels 1-13.
Try and get an Evolved Guardian Selection, and Advanced Summoning card as soon as you can. The Guardian that you want to summon is Griffin, who gives you additional armour each time it uses its ability. When Enhanced (has had two Griffin cards played without changing Guardian) this will be 5 each time, and 2 without. It will activate each time you play a Defend(Blue) type card, as long as you have Command, at the end of each turn, or whenever the card says "Activate the Guardian's Ability X times". These cards are your best friend, get them whenever you can, as they will activate the ability regardless of whether you have Command or not. Once you have built up a massive amount of Armour, switch over to the White Lion, and now the same cards that used to give you extra Armour now hurt the enemy for 3(7 if Enhanced) Damage per time, and activate whenever you have Command and play an Attack(Red) type card. Use the Armour you have to just tank hits while you pour on the damage with the Lion. This works for every enemy that cannot "Remove Defend", which takes away all the Armour you have. If you encounter a monster with this, still play your Blue cards, to mitigate the damage, but don't bother with the Griffin unless you have to. Just head straight to the Lion, and kill things as fast as you possibly can. Some enemies can also deal piercing damage that ignores Armour that you have, the same rule applies for these.

Levels 14 and 15
As before, we want to focus on Guardians to deal damage and help us stay alive. However, you cannot just build up a massive stack of Armour like you could before, as it shrinks by half at the end of every enemy turn. How we will deal with this is building up a stack of Armour at first, about three attacks worth with Griffin. Then, switch to Lion while focusing on playing Defend cards that activate the Lion's ability. This allows you to keep your Armour stocked up enough to survive the damage, but not too much as to waste Mana and lose it all to the halving, all while dealing damage with the Lion. A similar strategy is used with enemies that can Remove Defend. Get just enough Armour to take the hit, or only lose a few HP, then spend the rest of the turn dealing damage. If the enemy is going to hit you especially hard, switch back to Griffin and try and avoid as much of the damage as you can. Only have the Lion out when you are not taking any (or negligble) damage per round.

There are particular cards that are very useful, who's details are blow. Grab these if you can, in descending order of importance:

Improved Defence - 2 Armour. Activate the Guardian Ability. +1 Command.
Individual Training - 2 Armour. Activate the Guardian Ability
Soul Barrier - 3 Armour. Activate the Guardian Ability twice.
Training Camp - Activate the Guardian Ability whenever a Constant is played. Evolve this to give it the Promise attribute, so you can play it turn 1.
Master of the Guardians - Activate the Guardian Ability on the start of your turn, as well as the end.
Order - Activate the Guardian Ability. +1 Command.
Release Of Power - Deal 1 Damage to all enemies per card played this turn. Stackable if you have more than 1, but this rarely is the case. Evolved makes this cost 0 Mana, meaning more damage. (even better with Training Camp).
Prepared Guardian - Gain 1 Command at the start of every turn.
Beastify - 1x2 Damage. Activate the Guardian Ability twice.
Share Emotions - 3 Damage, Activate the Guardian Ability and +1 Command.
Getting Used to Loneliness - 1 Armour for every Mana spent.
Electromagnetic Field - Generate a random Constant card and add it to your hand each round. (this is higher if your deck already has Training Camp in).
Serenity - Gain 4 Armour for 0 Mana. Evolve it to make it usable twice per battle.
Switch Command - Guardians summoned are always enhanced, so you don't have to play the card twice for the better effects.
White Lion's Assault - +2 Command and instantly summon an Enhanced White Lion.
Griffin's Assault - +2 Command and instantly summon an Enhanced Griffin.
Reserved Force - Make 1 Random Card in your Hand cost 0 Mana, once per turn.

To add on to the two excellent guides above, if you find you're always struggling to beat the 3-5 boss stage, before selecting it, use the home button in the top right to return to the main menu. Afterwards, fully quit out the game and reload to the title screen and select "Continue". This will skip the 3-5 boss stage, and award you completion of the run.