Doh, I feel like such an idiot. Looking over the schematics for the TM241a and RC-20 manuals I see something I dismissed a long time ago. There's Serial In and Serial Out pins, but there's also a Serial CLK pin. Well, I was receiving data just fine without the clock, but apparently I've been spinning my wheels this whole time sending data to the transceiver. You have to clock the clock pin when sending data TO it. Unfortunately the Bus Pirate is apparently completely unable to clock it's clock pin in UART mode. Only in other modes like SPI and I2C. That sucks since I already have these nice probe cables for it and everything. I don't know if any generic FTDI type usb-serial chip does it. I think that's a pretty much dead part of the standard these days. I'd love to be proven wrong though.
Not even the latest v6.1 Bus Pirate firmware has support for this. You can see some commands in the help menu to twiddle the clock pin manually, but you get an error message in UART mode. :/