I just tried doing the stovetop soldering method described
in this comment thread.
I tried it twice with aluminum pans. Both of them started to smoke dry (residual cooking grime, I think) before the solder flowed. The fiberglass is a bit toasted, too. Not good.
Helpful Bella is still helpful.
This was fun, but I'm definitely making the next 20 of these with a laser cutter.