Since you didn't modify the original phone, I wonder if the oldschool dialing way of digit-banging would work. For each digit N, you hit the hangup button N times, pause, then go to next digit. Used to do that on our family's Pupin that looks like this one [1]
I remember it being N at least in ex-Yu. But now that I think about it I forget if it was a reverse-click i.e. the pulse was created by releasing the pressure plate briefly. In that case we'd have a fencepost thing and there would be N+1 physical presses maybe
[1] https://in.pinterest.com/pin/rare-vintage-pupin-bakelite-rot...