食べてみる means to eat and see, aka try eating, but can you do this:
食べないでみる meaning "I'm gonna try not eating"
今日お風呂に入らないでみたい is a valid sentence. Don't use kanji for 見たい since it's a subsidiary verb here. And it's お風呂に入る, not お風呂を入る.
Here are the three examples found on BCCWJ:
But the number of the examples was smaller than I expected, which may mean this construction is relatively uncommon.
食べてみる means to eat and see
In my experience, it used in the context of trying something to see if you like it. Someone might pass you something and say 食べてみて...
The negative form you mentioned, 食べないでみる, is strange in this context. Like saying "Don't try this and tell me what you think". However, in the context of some other purpose is OK as in "I think I'll not eat today to see if helps my stomachache"