Thanks for all the helpful suggestions!
1) I was able to use wire to secure the strike spring before removing the back plate. It's a fairly weak spring, so that wasn't a problem.
2) However, the time spring is much stronger. It is also a hole end, rather than a loop end, so the clearance when the hole slips over the barrel hook is minimal to non-existent. Since the "barrel" is stationary, the main wheel is attached to the arbor and a spring-winding collar cannot fit over both the main wheel and the spring. Sooooooo, I've removed the winding arbor, and will try using a "dummy" winding arbor with my spring winder to get the spring out. I think that the "barrel" is removable for cleaning purposes, but I'm not sure that it's sturdy enough to hold the spring without the extra support of the center plate.
I'll post some more pictures after I get the thing apart, so y'all can see what an interesting contraption this is...