I did the same swap, found a 1.7 cu ft that was nearly identical in dimension to the ice box that came in my '93.
I tapped the wires to the outlet just to the rear side of my opening in the base cabinet, and added my own duplex outlet with a switch to turn that specific outlet on and off. Figuring that there could be times when we camp, where I might not need to run the refrigerator. That new outlet and switch are inside the base cabinet next to the refrigerator, where I happened to have an access door. So it's out of sight, and takes a dedicated effort to turn it on or off, won't just get bumped accidentally. So I did basically the opposite of running a dedicated extension cord, I actually added a duplex outlet to the existing circuit. But I don't have much else pulling AC voltage, no air cond or furnace. I do run a oscillating fan most of the time, in effort to move inside air around so it doesn't get too stuffy in the camper. And we will use toaster, coffee maker with refrigerator running without issue.
A lot of people have concern about 'breathing space' around the refrigerator. I have mine set up squeezed in tight from top to bottom, and open to the rest of the inside of that base cabinet. And I have not had any issues, just a couple weeks ago I had daytime temps in low 90's and had 39 degrees in the refrigerator. I have a small brace that I constructed that fit between the rear of the refrigerator and the inside of the exterior wall to hold the unit in place. Have thin trim boards added to visible face of base cabinet, that fit in the gap where the door seal is, so they hold the 'box' in the cabinet and the door swings freely in front of them. It has moved a couple times when I have encountered especially bumpy roadways, but most trips it never needs any attention, it's ready to plug in and chill at the campsite.
Only time we have needed a separate dedicated extension is when we have 2 electric cooking items heating at the same time, or a space heater and one cooking item heating. Refrigerator current draw has not been an issue at all.
Good luck with the mod. Having lived with ours for a few seasons now, it is one of the best updates we have made.
1993 Jayco Eagle 8SD
DIY Curtains, DIY exterior awning, 120V compact refrigerator installed, thumping sound system added