How can one update a theme on a live site without interruption?

I think i’m going to answer my own question:

My problem #1 (no direct server access) is probably what makes this situation the most irritating; had that not been an issue, i think there would be a lot of additional options open to me.

But given that limitation, creating and uploading ‘new’ themes seems to be the best choice. Accepting that, the question becomes, how to solve the problem i mentioned in #2?

The answer seems to be:

This problem is a limitation of WordPress itself (tracker # 18588) that will likely be remedied in a future version. Until then, there is a programmatic work-around, provided by a user in the linked ticket (and also here on Stack Exchange), that can be used to transfer the settings between themes when the switch occurs. Having adapted that code to my own theme, i’ve found that the work-around is sufficient for my use case.

cheers