Replaced the Tee with a restricted one. It came from the part store ( Advance Auto ) and identically matches the the original, except it's not broken. Cost $9 and change.
Replaced the heater core with an aftermarket one, cost $41 and change.
Replaced the heater core hose clamps with regular old run of mill screw type clamps (already had them). The original clamps took over two hours to remove, due to the space restrictions around them and not having some secret Cadillac-only, factory trained, super mechanic tool or something. (ended up buying 11 inch pliers and a longer screw stick) That's roughly 1 hour per clamp. Rediculous, I know. I have replaced a handful of heater core's in my time, and I NEVER had such an issue with stupid hose clamps. By the time I was done with removing those evil clamps, I was effectionately refering to them as "bastard clamps." I have sternly scolded them and threatened to melt them down to nothing.
I decided to document the process by taking a few pictures of the operation. They don't do my torment justice. Actually, beyond the clamps, it was not that big of deal. But I know I'll never put a clamp like that in a hard to reach spot. Someone should find the engineer that came up with that idea and beat him/her about the head and shoulders.
So, thanks for the tips. I have heat now, or so it seems. It was in the low 70's when I started and upper 60's when I finished. As such, we'll see when Old Man Winter gets here if I have "good" heat or not. I suspect that it will be better than it was at the very least.