I seem to have fixed my problem.

Using Keychain Access, I made a backup of the com.apple.idms.appleid.prd.blah blah blah certificate. If this experiment fails -- i.e. if it turns out something else really needs the certificate -- then I can restore it to my keychain.

I tried BTMM to log into my home Mac, and the dialog box requesting the username and password finally appeared.

I get the impression I'm the only one on this board who regularly uses BTMM and worries about this sort of thing, but just in case someone else needs the information in the future, it's here.

Thanks, Steven and Gordon, for your help and attention.

-awlabrador