If you remove the doors from the refrigerator will that give you enough space to clear the island.

The side panel for the cabinet to fridge will be hard to remove. As suggested above if you remove the door will that give you enough room.

Shove two metal drywall knives between the crown molding and the cabinet, then drive a small wrecking bar between them to pry it off. Back the nails out please. Remove the cabinet doors and any screws affixing to adjacent cabinet faces. Place 1x blocking at the back of the cabinet face frame and smack the blocking until the face of the cabinet is removed. Go around gradually and it will pop right off. Pull the fasteners. Smack the fridge panel sidewall off and your fridge is free. Wheel in the new fridge and reassemble.

It's scary at first, but it ain't all that. Yes, you may crack the face frame but that's unlikely. It'll glue back together inconspicuously. I've done this thousands of times. Literally.

You do not need to touch the cabinets. Take the door off the fridge slide it out and take it away. BTW if that fridge cannot clear the island the island is too big and the walkway way too small.