Monday, with the battery freshly charged, I went to go fly. After all, if the problem is a slow leak to ground, then as long as the battery is charged enough to start, and the alternator is providing juice back into the system, the plane will fly fine - I just won't take her anywhere that shutting down for a while might lead to a problem.
That was the theory, anyway. The reality was that I pulled the starter, and I got the prop to turn over a little with an interesting grinding noise, before it stopped. This is not the battery - there's still plenty of juice in it. It's not a leak to ground, because that wouldn't keep the starter from going. I was wrong.
It's the starter. That horrid grinding sound before the prop stopped? Yeah, that's the gears not meshing. Thankfully, I didn't try long (about as long as it took to go Oh No That's Not Right). So today, tearing down the starter, and finding the problem starting from the engine back, while making sure the gears didn't leave any metal loose in the engine as they failed to mesh.
If it's the starter, which it likely is, I'm going to pull the engine, have the local engine shop modify it slightly to upgrade from 1941 technology to 1980's technology with a new Sky-Tec starter. They're smaller, much lighter, and far more reliable - so I won't have to deal with this again for many, many years.
And here's the part where I know God has an interesting sense of humor - modifying the engine, buying and installing the new starter, and reinstalling the engine on the airplane will cost... just a little more than the amount a friend just paid for my old radio, VOR head, and associated wiring. It'll be okay.
If you'll excuse me, I'll be out in the warm sunshine, working on my airplane, laughing about life and reminding myself that this happened before I was in the middle of nowhere over the Rockies, no one's shooting at me, and my husband loves me. All else can be handled easily.