Update:

Monday, a week ago, I shipped my MBA to Apple. It came back this morning. Same computer, shiny new battery. No other changes, so far as they said on the sheet that came back with it.

I had pre-emptively backed everything up twice (the usual over-the-network Time Machine as well as a Super Duper backup to a USB hard drive), and then I deleted a bunch of personal stuff and sent it in. Interestingly, the computer came back with the hard drive unmolested. Restoring it was an interesting adventure. The Super Duper drive turned out not to be bootable, even though it was supposed to be. Remote booting failed in an interesting way as well. The idea that you're supposed to hold down the option key when you boot. It then dumps you into a dialog where you can select which computer to boot from. Because we've got a big network here, it saw a ton of computers offering Mac boot media. It showed the first few of them and then stopped. There was no way to tell it to go to my specific desktop Mac with its backup. Maybe next time I'll use a cross-over cable.

Anyway, I instead restored my files the "normal" way, through the standard Time Machine user interface. The hardest part was getting Time Machine to recognize its old backup volume (along with a brief panic when it saw an empty backup, for some reason, and was about to overwrite it with something new). Once I sorted all that out, the restore did what appears to have been the right thing.

Tonight, I'll see if the new battery has any less drain when it's "off" than the old battery did...