So I was able to reset the tire pressure sensor within the cabin. I had to wait for the car to think it was OK again, stop, reset it, and then continue on. It lasted for about 10-15 miles before it crapped out again and all the errors came back.
So at this point I've come to the following conclusions:
I am in no way required to use runflats, despite the tireshop telling me so.
The sizes of the tires fit the stock requirements.
Resetting the TPM doesn't help.
Seems to me the answer is to take it to someone familiar with these cars and have them look it over and repair/reconnect whatever is messed up.
The problem with that is that it costs money, money I shouldn't have to pay for someone else's screw up. Taking it back to the people who screwed it up is useless because they don't know what they're doing anyway.
What would you guys do? Pay to have it fixed, then bitch at the tire shop and hope they reimburse you? That's looking like the only option at this point....