What to do with a gap in the countertop next to kitchen sink?
You can put a piece of wood on the edge, scoot the sink over by 1/8", and epoxy and caulk.  If this sink is highly utilized I wouldn't expect this to look pretty after six months.   You have the weight of dishes and water in the sink, you have dirt and grime, just nothing that will hold up like a countertop.

Even thinking of a weird solution that could cost you $10-20 to look crappy, a good sink can usually be had for $30-40 so I would personally install a bigger one.