Is there a way to escape mystery room fights in Disgaea 5?

This one looks like it's either bugged, or intentionally very evil.

The exit portal is there, right under the cat, you just can't see it but it works as normal.

As far as I've seen, there are no actual trap rooms in this one. You can always escape.

Furthermore, the bank girl is random. You can safely talk to her between 1 and 4 times before she summons backup, each with more money rewarded.

You can punch the cat off the exit portal. Triple Strike, or Hurricane Blow both work. It will still be pushed off even if your attack does no damage or even misses.

Both of the previous answers are half-right, but Omega, the room you are talking about I have come across. I did the same thing to end up finding myself fighting multiple level 9999 guardians at a low level.

This mystery room you cannot escape from. It may be 1 of a few (or the only one) that you cannot escape from, because they are trap rooms. Most rooms you can escape from by killing or knocking off the Slumber Cat from the exit portal and enter it. The only other option is to finish the room, usually by clearing the room of everything except the slumber cat.

All you can do in the trap rooms is remember what happens in each room and avoid them, or grow strong enough to kill the NPCs next time that happens. So far the only one I know of is the bank, and is only triggered if you talk to the NPC twice.