How would I hook up this dishwasher without going through the cabinet
between the dishwasher and sink?

First, consider these (anything else is likely going to be harder):

Option 1: The "floor" of the cabinet interior is elevated above the
floor of the room to the height of the toe-kick (the recess at the
lower front of the cabinet). Use a hole saw to cut a hole at the rear
of the cabinet below the cabinet floor and above the floor surface of
the room. (if you can only come at the dishwasher of the cabinet you
will need a drill extention and perhaps a right angle drill.

Option 2: If the cabinet has a drawer at the bottom, there *may* be
enough room to run the supply, drain and power behind the drawer if
it's secured to the bottom and rear of the cabinet.

Likely, you are required to pull a permit for the the plumbing, the
electrical, or both - the inspector(s) will probably like option 1
better than option 2.

Michael Thomas
Paragon Home Inspection. LLC
Chicago IL
mdtATparagoninspectsDOTcom
847-475-5668