Hey Tim,
On the surface I can't think of what would cause this unless JUpgrade is somehow also pulling the tables #__bsms... into your new database.
I'd check right after the upgrade but before you run the migration component for JBS to see if there are any of those tables pulled over.
I admit I haven't yet used JUpgrade on a site and migrated JBS so maybe there is something I'm missing.
Just to go over it again - the steps are:
1. Run JUpgrade on your old site, which creates a new db and a new folder for your new site
2. Run the JBS migration component on your OLD site's db - backing it up
3. Run the JBS migration component on your NEW site, importing and migrating the JBS db you backed up
4. Install JBS 7.0.0 on the NEW site.
Check the db to see if you have #__bsms... tables at each of these steps - that might give a clue.
All else fails - you can open a support ticket (link at bottom of the page) and give me admin access and I'll take a look. I might also need access to your phpmyadmin. One way or another, we'll get you migrated over!
Tom