That seems to have made it work, thanks, but it was odd.
I uninstalled Locus Free/Beta, Ran Locus Pro, asked for the Live Map. It said "Enabled" but nothing happened. I tried it a few times; noticed it said a mix of "Enabled" and "You need to sign in" or words to that effect. So I signed in (took a very long time, but I think that's Groundspeak's fault), and now it's working fine.
Like you say, maybe it was some confusion between G4L and (Pro vs Beta). And as I said, I rarely use the live map, so if it's only me mentioning it, I wouldn't worry about it.