Is everything miked and running through a house mixing board? Overdub a capture of that house mix with your video. I'm sure I'm preaching to the choir here. You seem experienced enough to know that yourself.
When I used to gig a little we'd always have the house mix plugged into a laptop which was capturing everything. In fact we always have it capturing separate tracks of each instrument. That way we could adjust things later if we wanted to. They came in handy a few times. Most of the time we just ditched the files after a while.