Iron core transformers may not be all that unavailable. Keep in mind 3rd world countries get all our junk. Which includes tonnes of wall warts that are too heavy for our tastes anymore.
I've built that before, which is also basically a commercial unit (which you might rightly ask, isn't very expensive either so why not just buy one and have it shipped in? -- but therein lies the problem, what if you can't even do that?).

Shame I didn't write down the schematic, I don't remember what all it does. Bottom-center: TL494 control board (with opto feedback, I think), bottom left: push-pull forward converter (think there's a dual gate driver on the board). Current mode control. Also a UC3843 flyback converter for HV side supply; this is a fully isolated design. Transformer is good for about 200W, and runs at 100-some kHz I think. FWB output and choke filter. Uses SiC schottky because I can (they run very cool; you'll have to be mindful of heat dissipation using FR307s or whatever).
Power (170V DC) crosses to the inverter board on the right, a TL494 at ~120Hz and fixed (~50%) duty, and, I forget what I did for current limiting but there is active current limiting at the output. Chokes on the breadboard are for output filtering, and I later added some film caps to filter things further.
Output: "modified sine" (50% pulsed bipolar) waveform, correct peak and RMS but dV/dt obviously wrong. Current limiting gives excellent performance into capacitive loads and short circuits (I didn't test shorts, come to think of it, but it should be okay for at least a little while..).
But as you can see, it takes a lot of parts to do this, and a heck of a lot less to use an iron core transformer.
OP: would it not also be attractive to use BJTs instead of MOSFETs? A discrete circuit could be made, without needing anything more than C1815 and A945 (or any other GP BJTs), some power transistors, and which drives a 12V winding H-bridge style, 100% square wave, without current limiting. (And could add current limiting for probably less than a dozen additional components.)
Tim