Okay, so get this. The Farm and Fleet guy had gone around to all four tires and tried to link them up with the TPMS--he did that several times. His final answer was "derp, maybe it will just go off on its own."
I read other suggestions that it had to be driven at a certain speed for a certain amount of time, etc.
Well we drove it around with the light blinking for about a week and then I had to go out of town for work for a week. The following week it was still blinking all the time.
After, well, how long has this been? Maybe four or five days ago I started the car, drove about two miles, the blinking stopped and the TPMS light came on. Then after a few moments it went off, maybe blinked once or twice, and then shut off.
And neither the TPMS light nor the blinking tire light have been on since. So I guess my old mechanic dude was right and I just needed to drive it a while?
Either way, it finally went off, so I'm happy. This weekend was going to be the weekend of driving back to Farm and Fleet and arguing with a manager about how they were going to be paying for me to go to Honda to get something fixed that was working fine until they replaced my tires. Looks like I don't have to have that conversation now, so yay for me.
Thanks for the replies, folks!