Since you are designing the whole house, you could steal a little space from the adjoining room to push the fridge back some. That way it will be even with the counters and not intrude out past them.

I also have a large island in an L shaped kitchen (last summer remodel). We don't have anything on the island. Strictly for chopping and two sides hold stools. We have the coffee station at one end of the L, then some counter space, then the window and sink. Blender in the turn of the L then the stove/cooktop, then the fridge at the other end. It works well for us.

We just finalized (well maybe!) our kitchen layout. It will be an L with an island also. I decided not to put anything in the island because I don't like all the pivoting that is required when you have to go back and forth between sink, cooktop, refrig. Think about how you like to move around in your kitchen. It bothers me to have to walk more than a few steps with bags of vegs to the sink. I personally don't want a hike. Ask yourself what will be easy and desirable for you. Sometimes due to limitations in a space, we have to sacrifice something. The trick is to zoom in on what sacrifices you can live with (it is your kitchen).

You say if you place the refrig on the other side, it will make your counter space very tight. Perhaps but will that be compensated by the counter space you will have on your island?

I hope this helps - these choices are hard! :)