"Makeshift flamethrower" or "handmade flamethrower"?
Handmade is anything made by hand:-
made by hand, rather than by machine [Random House Dictionary]
But you could have spent weeks assembling a flamethrower with tender loving care and it would still be handmade.
Makeshift is something pressed into service on the spur of the moment:-
a temporary expedient or substitute [Random House Dictionary]
So for your purpose, makeshift would be better. You could also try improvised:-
to make, provide, or arrange from whatever materials are readily available [Random House Dictionary]