I'd find a local granite fabricator. There, you can select a remnant of sufficient size to fit with the necessary overhang. And, they can cut backsplash for the back and side(s) from the same piece.

Talk with them. It might be necessary to let them see your space and/or the faucet and basin. They will cut all necessary holes and probably install the basin.

It will be surprisingly inexpensive and look finished.

Buy a 1/2" x 34" piece of white solid surface from a fabricator. Install a cleat on the wall then the vanity top with 1 1/2" overhang and silicone the solid surface strip over the gap in the back. This will be much less expensive than having a granite fabricator make a new top.

We had a bathroom vanity top cut from a remnant of granite. We picked it up and installed ourselves. It was 55" and ended up only $25 more than laminate. Check into it.

We had a similar issue with a powder room vanity in an alcove, the granite was cut 1/2" too short so there was a gap with the back wall. Our GC suggested putting a new piece of drywall on the back wall to close the gap. We didn't do this (a granite backsplash was deep enough to close the gap) but maybe it would work for you. Also, there are a lot of tile trim pieces that are pretty thick so maybe you could find something that would cover the gap.