I've never met a bathroom that didn't have room for a trash can next to the toilet. I wouldn't wreck my counter to get the job done.
PROI agree, I don't think that you will achieve the look that you are going for. I would find a really nice waste basket that you can leave out, or put under the sink.
I get that you want a handy trash basket. Do you have a closet? A vanity with doors? Put the trash can in one of those locations. If it's a bother to open and close the vanity door each time you need to dispose of a Q-tip then put it in easy reach while you're doing your thing and then back out of view when you exit the room.
What about making one of your cabinet doors either slide out (like a kitchen trash pull-out cabinet) or one that tilts out with your trash can? It sounds like you have room to do so :)
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