Stainless steel binding wire. Get the heavy stuff. (Rio, Otto Frei,
pretty much any supplier will have it)
If you get the thinner wire (26 ga (ish)) twist it into a double
strand. It’ll work better, and won’t burn thru as easily.
Alternately, you could try tinning the solder onto the ends of the
tubes, so the solder’s already melted onto the tube ends when you go
to solder. Neater that way.
The biggest issue is learning to heat gently until the water in the
flux has boiled away. If you heat until you just see the flux
start to bubble, then take the torch away for a second, and keep
repeating that cycle until it doesn’t boil any more, you’ll have
much less trouble with the Mexican Jumping Solder. It’s actually
quicker, once you count the time and effort of re-positioning solder
that’s jumped into the wrong spot, or parts that have shifted.