Maybe I've inhaled too much smoke in my life - no I definitely have. But, if heat was more important than esthetics, I would fire that thing and see what happens. Obviously, don't breathe the smoke. If it was in a place I could establish some serious, fan forced ventilation, and I could lay a decent fire that I could light and leave (depends on the stove), that's what I'd do. If that went okay, I'd do it a couple of more times and see if it stopped smoking / smelling. It's best to work your way up to "hotter than you'll ever need", just so it gets the smell thing out of it's system before it's the coldest day of the year and a problem to get fresh air.
Even with only stove paint, the above is exactly what I do anyway. The stove paint fumes give me a headache, and smell awful. I just burned the dust off of one of my stoves today, and I arranged to be mostly not present until the air cleared.
I looked at the MSDS of Rust-oleum Chalk Board paint, if that's what you have, and the hazardous products of decomposition are CO, CO2, and maybe NOx. Same as the stove and BBQ paint MSDS I looked at. Meh, wood has all those and more when you burn it. Do your own research based on exactly what you have.
Once cooked, you have to decide how you like the finish. If the top coat fails and chalks (pun intended), you might be able to wipe it off with a wet rag, or a step further - wet fine scotchbrite. If you have to go farther, you might find that once cooked, you can sand it some and recoat with stove paint with good results.
I guess if you wanted to get all scientific about it, you could paint a metal scrap with the process you used, simulate firing it and see what happens. If you don't like the results, you can then experiment with the fixes.
Personally, I'd risk making the whole thing maybe harder to strip, for the chance that I'll be fine once fired. Stripping and repainting from scratch is going to be a messy, smelly, labor intensive project. Firing it is just smelly, regardless of the paint used.
If it has to look perfect, I have no suggestions. I know a guy who prepped and painted a Jeep frame with pro urethane paint. It looked way over the top, for a daily driver vehicle, to me. He decided it didn't look as good as he wanted, so he stripped it and did it over - months. Different strokes!