After discovering that there was no official way to sync my iPhone to multiple iTunes libraries, I did some searching, and found a workaround. (I did try the method suggested on many sites, such as this one, but I wasn't able to get it to work.)
While poking around, I inadvertently discovered that if you check (or uncheck) Encrypt iPhone Backup (on the iPhone's Summary tab in iTunes) that this forces iTunes to create a new backup of the iPhone (after you are prompted for password) -- even if it's not a phone that is normally synced with that library.
While I suppose that this could have certain security risks, it gives me the ability to force a backup. Of course it's a full backup, so it takes a few minutes, unlike the normal sequential backups that happen with a normal sync.
This is especially handy right now, since I'm away from home and potentially taking my iPhone to have service done (battery problem)...and the first thing they ask you is 'Have you backed up recently?' Should anything bad happen, I want to be sure I'm covered. And now, I think I am!

