Whether a 2032 cell works depends on the circuit, but most likely yes. It just won't last as long as the 3.6V lithium cell.
I wouldn't use an 18650 Li-Ion, since the maximum voltage is too high, and you'd have to take it out frequently to recharge. (And you absolutely must use a cell with built-in protection IC that'll stop it from getting deep discharged.)
What's wrong with getting another 3.6V lithium cell? They're not particularly hard to get (by mail) and last forever with a very stable voltage until the very end, which is why they're used for these applications.