I did something similar for my home lab setup in my previous house. It was pretty reasonable having two dedicated 20A circuits run w/ surge protected hospital-grade outlets and dual function breakers. Each circuit fed a different UPS which fed a different PDU so everything had redundant power back to the breaker panel, which was all I could reasonably do residentially, and it meant none of the servers/network gear impacted the rest of my office circuits.

It was reasonably cheap, and in my next house I'll do the same again. Running additional circuits is pretty easy if you have an attic or crawl space.

Are you exhausting the gaming computer to the outside or into the room you are cooling?

Into the room I'm cooling. I suppose it would be possible in theory to do so, but the particular layout of the room makes it difficult to impossible to exhaust both the AC and computer, I think.

In previous heat waves I've seriously considered venting my PC through the wall straight to the outside, but alas I currently rent.

Do you have an inroom AC unit? You could run a dryer hose from the back of your computer to the same window vent.