It sounds like your plan is to replace all cabinets? If that's the case submit a floor plan showing all dimensions including doors, windows, entrances to the room, etc. Also show the location of your existing appliances and sink. I looked up our Idea Book for an indication of what you like style wise but you don't show anything so you need to guide us for that as well.

yes..skooch fridge to the left....install 15 or 18 inch deep counters between fridge and dining room passage.....is that what you mean by full depth wont work there? once kitchen desk and chair are removed.....take th equivalent of work island [all drawers] and install it as a peninsula coming out from desk chair spot....so you have two L sides of the kitchen ..fridge will be centered better and more sq footage in the middle for foot traffic and general movement. Provide measurements.

just verify, you're replace all of the cabinets? island too?

getting rid of the desk area and only getting base cabs?? not a pantry?

I know you say you have a big family, but a counter depth fridge is the only way to reduce that depth into the room. I did it with mine. perhaps you could get a smaller, cheaper fridge to keep in the garage to store items. I bought a used one for $200 and have done that for years.

YOu also need a fridge surround if the sides will be open. I'd do full pantry w/a counter depth fridge. you could also do a built in micro like shown here.

a diff layout

if you want to move the fridge on the other side of the sink: (ignore the shape of this island)

or

I moved mine to the middle - great solution.

Only counter depth would fit in the middle. Two units one fridge one freezer? If they make them counter depth?

Island is too close.*

Look at the room behind the fridge. Can you recess the fridge into that space?