I wouldn’t check in late at all, and just have the later group check in when you do. The worst that can happen is they’ll say “come back in 10 minutes”.
They recommend that you check in 15 minutes early - if they can seat you then, they will, if not, you'll have to wait. I see no reason why everyone doesn't try checking in at 12:45 pm. If for some reason the later reservation doesn't get seated with you at that time, they'll be seated shortly - find a big table and wait for them (and eat slowly!)...