Easier than it sounds.
For every floor boss, get a long duration sword skill ready (Nightmare Rain, Eclipse, Starburst Stream, Nebularaid Empress are all great) and some form of invulnerabilty (PRT Armor). Once the boss gets down to critical health, order the AI to fall back, use your invulnerability skill and then take it out in one skill.
Note; even if the boss is dead, the sword skill will continue landing hits. The AI won't attack after the boss dies, so this is why a long duration skill basically guarantees the LAB for you.
If you don't get it, quit out on the results screen and try again. The game doesn't save until you get back to town, so it's safe before then.
The Vita version had a bug where you needed all the items in your inventory for it to work; I kept most of mine in storage and had a couple equipped, and it popped just fine, so that should be safe. I'm not sure if selling them breaks it, so I'd advise just storing the items instead.
If you managed to get them all, the achievement will trigger on the results screen of Floor 99's boss.
Specific notes;
Floor 98 -
Floor 99 -