So here's how it shook out:

I managed to find a suitable switch at Home Depot (I tried Radio Shack
first, but the clerk seemed to think I was crazy coming to Radio Shack
to buy an electronics component), and tried to wire it up.

It was fairly easy to get the switch connected, but in the process, I
knocked all the speaker lines loose.

So I popped out the radio and reconnected them.  But in doing so, the
lines to the antenna came out, so I hjad to take it all apart again.
THis time, I connected all the wires with scotch clamps, and clipped
on an extension to the antenna lines. THen I drilled a hole in the
lower part of the dash and mounted the pushbutton switch.  It's in a
slightly awkward place, but it was the highest I could reach without
disassembling the entire dashboard.

Unfortunatel,y the process of making sure all the speaker wires were
still connected required that I leave the radio on for quite a while,
and the battery went flat.

A jumpstart later, and everything worked. I can now lower the antenna
manually by pushing the button.