Who built the island? What's the concern exactly? I assume it's that the weight of the counter will cause the floor of the cabinet to tear out? If so, then the structure may be able to be modified to reinforce the floor/wall joint, or the caster can be brought to the edges and mount to the bottom of the walls in addition to the floor. That would put the casters all the way out on the edge.

If the island is secured and there is at least 36" clearance on all sides, your appliances would be able to be serviced/replaced without having to move the island.

It doesn't meet needed aisle clearances as is. With casters on it it has the potential to tip since it will be top heavy and not fastened to the floor. You've painted yourself into a corner literally. The kitchen is too small for this island. It needed to be a smaller, lighter, cart. Remake it smaller. Without the stone top that makes it top heavy. Or with a counterweight in the bottom to change the center of gravity.