Well, I've not really solved the problem, but I've ameliorated it somewhat. Two of the tower computers have been moved down to the basement. Only computer left is the work laptop.
I tried a couple of "over IP" HDMI KVM extenders, and while they kinda worked, it just wasn't good. I do have a gigabit network, but I think those guys need multi-gig networking or possibly consumer grade is insufficient.
So I ran a pair of CAT8 home runs between the basement and office—out the basement wall, up the side of the house (partly tucked behind a gutter downspout) and across a bit, into the attic, and down into the office wall to a keystone jack. The basement end of the cable just got plugged directly into the router there.
These are the extenders I used:
The first one I kept even though it didn't really do "over IP" well, because it otherwise worked well over direct cable and it had an audio port, and does 4K which I needed. The second one I bought because it was inexpensive and I didn't need 4K for the other computer. (I'd bought a third when I was trying to get it to work "over IP" but that one had an annoying idle screen that wouldn't go away when the monitor was supposed to turn off, so it got returned even though I eventually decided to go direct cable—what's the point of having an idle screen?)
Temps are a little cooler now, but there's still a degree or two of temp differential between inside and outside the office. I may revisit additional insulation at a later date, but I can live with this for now.