Can't fit the "whatever" in the cab because the opening is too narrow? When you have the exact measurements (cabinet opening without door) in hand contact Rockler.com. They have "offset" hinges which may solve your problem. OR, consider changing the door to one that lifts up. This option will take up "side space" within the cabinet. OR, skip the door entirely - that would be my choice so as to show-off your beautiful countertop material.

like idea to skip door

IMO skip the appliance garage completely .There is a way to have a lift up door that slides inside above the appliances but I don’t know if there is space in there. These are a couple of ideas

A piano hinge? Really. Of course there are other hinges that allow more motion. Like the standard Blum 170 ones used on lazy Susan’s. But the cabinet may need to be larger to allow for the clearance required for those type of Euro hinges. It depends on what size clear opening is required.