I do see some valid criticisms here, thank you. Heating with electricity sourced from the car is out. I am familiar with Ohm’s Law, after 44 years of operating hydroelectric powerhouses.

Running an electric heater from a small car’s charging system is not realistic. I do have a 12v “rear heater” somewhere in my junk, acquired years ago but never installed after I saw its current draw; it came with a 30 amp fuse and heavy wiring.
I understand the concern about leaking coolant. I’m sure nearly all of us are familiar with air hose quick connect fittings where the female end seals but the male end dumps air. But I’m not sure everyone is familiar with hydraulic quick connects that seal on both fittings (if you have a tractor with loader or backhoe, you probably are). I was thinking of something like that. Teeing into the heater hoses and running hose lines underneath, to the back of the car, is not rocket science. Adding a bit of closed cell foam pipe insulation would be easy. Still, it is a significant modification to the tow vehicle, so when you change vehicles...
I recall a 1966 trip with my folks in a 1964 VW camper that had an onboard gasoline furnace. I doubt I’d find one of those today.

I’m thinking of a heater I can run when I’m not in the trailer, but driving down the highway. A small propane catalytic heater or even an Aladdin lamp (equal to a 60w bulb for light, but equal to a 750w heater in thermal output) would do the trick when we are in the trailer (we’d be sleeping in Wiggy’s bags, so the heater will be only a convenience when we’re awake), so long as we remember to crack a window or vent. And, except for a tiny 12v fan on the heater core, it’s “free” energy (yeah I know, TANSTAAFL, but it’s heat we’re throwing away).
I want a very light trailer, and a spare heater core from my junk collection wouldn’t weigh much. I will probably have a better idea about feasibility after I nail down more details about the trailer, and spend some time under the tow vehicle (9th generation Corolla, manual trans). We know we want light, aerodynamic, and simple. No plumbing, so long as I can put a 5 gal water jug with spigot on an upper galley shelf when we stop. No A/C, but possibly a hatch where I could slip in a small window mount if we knew we’d be camping in hot weather and near 120vac power. My wife did say she wants decent insulation for cold weather.
I’ll kick the idea around as I learn more about tiny trailers.
Life is too short not to be happy.