remove the vinyl/underlayment. if the subfloor is wood, use a underlayment made for ceramic tile, like: hardibacker schluter ditra durock wonderboard and do not use type 1 or type 2 mastic. use latex-modified thin set mortar.

We are doing master bath remodel and previous owners had vinyl plank flooring laid over the concrete foundation-- the people who did the work said that if the baseboard is high enough (ours is about 3" of wood--not tile.. they said the vinyl plank was run to the baseboards and cut to fit then silicone caulk was used along that joint... So far have noticed no issues anyplace in house this was done--not just in bathroom... we are taking out all LVT in master suite and replacing it with 1 type of flooring for unified look after master bath is redone--since we can't get the type of flooring in there now--not made anymore-- they won't need to add a quarter round on top of the LVT... so as long as you are happy with how the LVT would look butting up to the ceramic tile baseboard--maybe you can do the same... IS the ceramic baseboard in good shape? Still tight?

I think you would ruin the ceramic tile for good. For the next remodeling it would be a matter of removing tile and planking. R u concerned about removal cost? Those guys do that in a blink, so not a big budget component. Yes, it is messy and you'll be dusting for weeks. Even so, my professional tile setter sprinkled water when removing old tile which reduced a lot of the flying dust. You could do it yourself to save $ but the time and effort are not worth it, IMO Another consideration is that when you leave tile and add something on top, you raise floor height and will have to trim doors, etc. It is not a cost saving strategy.

With lots of expensive prepwork, and undercutting all the casings and doors, and losing height of the baseboards, yes, you can halfass the project and lay plastic over tile. It won’t be any cheaper than doing the project correctly and removing the tile. It’s likely to be more expensive, actually.