It should be entirely possible if you don't lose to any actual battles on the way up, though it's definitely not obvious. If you don't have previous saves before hitting the 5+ loss count, then not much you can do without save editing/cheat engine abuse/etc. (If you do have this ability and want to just do that, set variable 62 to 0).
If you lost to Azusa Mifuyu way back in the hunter training school, then you'll be forced to lose to her in order to proceed through floor 2. (1)
You're forced into 3 losses in order to proceed past floor 5 (in order to get the key)... also, if you don't have the 'Dungeon escape' spell by that point, you can't get out anyway since you can't go down from floor 5 (mapping error in original and english patch only worked with text, not any map issues of which there are a few). (4 total)
And on floor 9, you're seemingly forced to lose at least once in order to pass by the 2nd npc there so that you can pass through her - any 'loss' to her should let you pass through her afterwards (5+ total, which guarantees loss to Melissa, as the status effects absolutely cripple you and will ignore any immunities you ought to have.)
So if you didn't lose to Mifuyu, you should be able to just take a 1 loss to the F9 npc, assuming you didn't lose anywhere else you weren't outright required to... which would have you at 4 losses, which is low enough to have a chance. But if you did, there's still a way... on the tile directly to the right of the offending F9 npc, face up and use your interact key and you should jump the gap (like I said, not an obvious thing - you can jump back to that tile from the one you land on by facing down, as well).
As for Asuka Ryujo, I think she only applies debuffs if you don't try to go on the offensive initially - the battle is probably winnable normally if you're so inclined, though it'd be extremely drawn out as I'm pretty sure she has the maximum HP (999999) the game allows for an enemy on either encounter variation (the bj version will however give you debuffs that should make winning legitimately impossible). However, you can potentially win in a single turn using incubus glare, as it turns out that (unless I'm misremembering) NO enemy in the tower (including all the bosses) is immune to it - I remember putting on Kaede Akino's training belt for that battle, actually, since it doubles XP gain and halves MP costs, despite the massive stat drop while equipped... you can't survive long with it on even near max level, but you shouldn't need to either if using incubus glare (though it's definitely possible to whiff on it working a few times in a row if unlucky). The book for learning it should be available from a certain NPC in Yapu town after dealing with the guild mission in Sodom dealing with a certain dungeon that appears right outside Sodom.
...on a bit of a side note, incubus glare is also my recommended method for taking out Katarina, should you try to grind her via save/load to get her special (1/200) drop, only available on the hardest version of her (after feeding her 4+ times); while she doesn't have massive HP, you would be pretty heavily debuffed in that version of the fight (and she can be OHKO'd by it, so...).
In any case, after Melissa is down, there's nothing really stopping you from losing as much as you want in the tower*, though I'd still just say to have a save ready before doing so anyway.
*Assassin enemies and a few others will give game over on loss regardless, and anything that doesn't is liable to drain level/stats from you without any direct indication they did so... levels can be regained easily enough, but I imagine you realize that stat up items aren't exactly the most common thing to find legitimately.
...oh, and one more side note; there's a few events you can do in the throne room before the king is forcibly teleported back there... just go there at any time after the tower is open but before he's back - one from sitting on his throne, another from the madame in the other throne, and a third from the maid after the madame's (I think this one should be repeatable after the king is back if it's unlocked before he is back).