There is an outlet against the wall that is on the other end of the island. Couldn't I make that a GFCI? Otherwise I would have to mount one inside a cabinet as there will be no aesthetically pleasing place otherwise.

Provided there's no cooktop or sink, an island (or a peninsula which it seems that you have) only requires a single receptacle. The receptacle must be GFCI protected (and should be already, remember that a GFCI CAN protect downstream receptacles).

You could replace the GFCI receptacles with regular ones, and then put a GFCI breaker in the breaker box to protect the whole circuit. This would be a way to have ready access if it trips.

"There is an outlet against the wall that is on the other end of the island."

Islands are not against walls.

An outlet not ON the island itself does not meet the requirements.

In our case you can enter the kitchen through passages on either side of the island and there is a wall just wider than the "island" at one end. So I guess technically it is a peninsula but it's more like an island with a wall on one end!?

The NEC requires 2 small appliance ciruits, but even with that requirement, I prefer to specify a dedicated circuit for any microwave or high wattage heating appliance like an instant hot water. Microwaves draw a lot of amps, and even a 20 amp appliance circuit can trip the breaker if you're running the microwave and maybe a crockpot or rice cooker, etc. If you aren't confident enough to run wire from the box, then call an electrician to do it.