Thank you all for the positive comments and the hints as to where to look for the problem.
I was convinced that the problem lied in my six year old battery. I took the battery to my local NAPA, where they tested the battery for several characteristics. While the battery isn't great, it isn't shot, either.
The negative battery connector didn't look the healthiest, and it had been cut off and replaced once too often. I bought a new cable, cleaned the ground area, added dielectric compound to the joints, and the 1250 came to life as it should.
You'll see it in the Potato Days parade on Saturday.
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