The problem... resolved itself.
I'd been wrestling with this issue since Friday, including a couple of hours today. I rebooted the laptop, restarted SSMS countless times, etc, etc.
While on the phone with support (of course!), I decided to right click on one of the tables (in SSMS) and select "Design" immediately after the SSC timed out on me. And, right after the table design grid came up, SSC updated with all the tables that needed to be checked into SVN.
Now there is no timeout, even if I restart SSMS. Go figure.