Well, I had my plumber Brother-In-Law over this weekend and we were able to partially solve the problem. It turns out that there was a Backflow Preventer installed in the drain already but the stopper had broken off. I replaced it for $20 and about 30 seconds work.
This solves to problem of sewage back-up but there’s still a chance of the weeping tile back up into the basement. I’m not too worried about that because to problems in the past have been back-up from the sewer and at least if it’s from the weeping tile the water’s clean. Now it’s just a matter of addressing the leaks.