Hello,
Yes install sink in the island and now you can leave frig there or on wall where sink was.

36” aisles aren’t big enough... your kitchen cannot fit an island with this layout, and a prep sink does nothing to solve your dilemma. (Also, you want to figure zones, not the dated ”work triangle” idea.)

If you post a diagram of the entire floor, we may be able to come up with something different.

An actual to scale floor plan empty but show windows doorways and all measurements clearly marked . If plumbing can’t be moved make sure you state that. How many people cook in the space on a regualr basis? No to any small walkways anywhere for sure. Also at least mark what spaces are next to the kitchen

36” aisles are too narrow....42” is my minimum, with 48” preferred. I like the fridge and the sink closer together. The range can be further from the sink. The island can be narrower. Seating at the island is a bad use of space if you have a breakfast room/area nearby. Plumbing is movable, but the cost of moving it can skyrocket if your home is on a concrete slab rather than a raised crawl space. Keeping plumbing where it is saves money.