I'll bet it would be regulated somehow
First - - I have no personal knowledge of the specific law as applies to a flamethrower.
Second - - I'm not saying this is the way things SHOULD be.
But, there was a force-fit decision made some years back, long pre-11 SEP, that BATF had some jurisdiction to investigate crimes of arson, where an accelerating agent was used- - gasoline, that sort of thing - - from the standpoint that anything that
COULD explode,
could be a bomb. I don't know at what point they decided that non-accelerant arson was an ATF matter. Maybe it technically is not. But, if they're called in to assist in a local arson investigation, it's kinda like the camel getting his nose in the tent. Nowadays, of course, there's the possible terrorism aspect, which can automatically get just about any ol' federal agency involved.
What does this all have to do with a military-type flamethrower, one asks? Well, there's the accelerant aspect. AND, what's a flamethrower FOR, except to commit arson - - I mean, setting stuff afire, forgetting some local requirements that it be mortgaged or insured property in order to qualify. Yeah, I think: Any Other Weapon.
Re-reading the above, I find that the reasoning looks a bit convoluted, if not a little confused. And if
I feel that way, I wonder how much sense it could make to someone else.
As the late Rosanne Rosannadanna might have said,
"NEVER mind."
I'll not delete my blathering, though, in case it sparks a train of thought in someone else's mind.