That is something I'm not sure about; it wouldn't be something theme-specific. That's more of a general WordPress question.
One thing you might want to consider is turning the main site into a WordPress multisite installation; that way if users log in to 1 site, they are already logged in to the other. It would also simplify the update process - A multisite installation runs (as the name suggests) multiple sites from a single installation of WP; if something needs to be updated, you do it once, and because all the sites in the "network" are running from the same installation, the single update covers all sites.
Using the Multisite/Network method, you could just create a "members" user group and limit access to the second site so that only the "members" group (and admins, of course) can view it.
As I said though, that's more of a general WordPress question, and not something I have first-hand experience with; I use multisite a lot, but not for membership sites or with the login scenario you are suggesting. You may be better off checking out the WordPress support forums
to see if anyone else has done something similar.