That is not enough information to tell you much. Even a picture would have helped. To start with, you need to know how many cables are going to be in the box. If it is one 12-2 cable + ground, then you need a box with a volume marked on it that is greater than 11.25 cubic inches. You do not say how much room there is between the outside of the cabinet and the drawer wall. There is plenty of room for the right box on a standard drawer cabinet. You do not say how much room there is between the back of the drawer and the back of the cabinet. You do not say what the total configuration of the island is, or where/if there is an overhang/bar etc. It would be nice if you could post the cabinet plan and model/brand of cabinets.

Tombstones on the countertop will work. If you don't have much of an overhang you can usually find some location that will work and not interfere with the drawers. You can build out the face of the cabinet a little if the clearance is close.

I'm an electrician and use these in a pinch and they are tamper resistant as well , obviously run it by your local inspector but it should be fine

*as long as you don't try to install them face-up in the countertop.