I would remove the drywall about 12" up from the vanity top, shim out the bowed area and install a piece of 1/2" cement board. That way you've got a good flat base for tiling the back splash. Since you won't see the area behind the mirror anyway, the seam where the cement board meets the drywall can remain unfinished where it doesn't show. Where it does show you can finish with regular drywall mud, sand, and repaint.
An alternative approach could be to float out the bowed area with hot mud (powder mix drywall mud). This will take some finesse and as a tile base probably not ideal but I think for a small backsplash it would be fine.
I have to do these kind of fixes sometimes when trimming out (carpentry) jobs where walls are wavy and there is a gap too big to hide any other way.
Them framers and rockers just figure its up to the trimmers to make it all work.