I'm no expert but once the babies are grown up, they should move out. They should only be returning to 'nest and mate'. So, if you wait 'till Fall, you should be OK in screening up the entrance. Do it early in the day when they're out feeding. If you can find out where they're actually building their nest, you'll know for sure if they're all out. As for having multiple entrances, that's possible. In the Fall, plug one, then eventually the other. Again, do it when they're out feeding.
As far as the house being huge and having 3 levels, I don't know what to say other then buy/borrow a ladder. My best to ya and hope this helps.
PS: Maybe you can 'rig' a one-way door at the entrances. Talk to a friend who is 'creative' or imaginative.