You may not have realized it, but vinyl is usually installed over a 1/4" underlayment which can be easily removed, and it should be. Double check that's what you've got. You can have the 1/4" & vinyl removed in a short time.
That's the best way. As you mentioned the vinyl has curled near the tub, guess why? Right, because of water. That means the 1/4" ply may be in bad shape.
Having said that, sometimes leaving everything can work. But, most vinyl sheet flooring is cushioned. It's not real soft, but cushioned still. Removing the old will save you over 1/4" in height too.
Please do NOT
install tarpaper as someone mentioned.
There is no approved method by any manufacturer or association to do that. If you need more specific instructions just ask.