Can you get the same cabinets? The counter top would have to be replaced and I think using the DW in the peninsula would be awkward

Maybe instead of a peninsula you can do a small island with just two cabinets? This way you can get away with not matching cabinets and counter tops. It's "in" to have different colored islands and island tops

The process of doing that is pretty simple. The most complicated part would be running the water supply and the drain around the "corner" to the new location.

I'm not sure that I'd want to be twisting around every time I load the dishwasher though.

IMO there is not space to do that ans still have a functioning space at the sink but I would need to see a floor plan done to scale but usually the best place for the DW is beside the sink.There is for sure no space for an island.

To answer your questions - it probably isn't difficult, but I don't think you'll like the results. It would be very awkward to try to rinse at the sink and load the dishwasher. Plus - think about where you are storing your dishes, and how far away from the dishwasher they would be. The expenses: plumbing, matching cabinetry, new countertops.