OK, I have solved the problem, though I didn't see any mention of this in the data sheet.
In order to determine when the unit goes from TX state to IDLE state, I was reading the MARCSTATE register while waiting for the transmit to complete. For some reason, this puts the CC1150 back into TX state...and of course with no data in the FIFO, it started sending preamble bytes.
The fix is to instead read the chip status byte with an SNOP strobe. It seems to be safe to do this during the transmission. To be extra safe, I wait for several msec before I begin polling the status byte, since the data sheet says the status byte might indicate IDLE when in "some transitional states" (including CALIBRATE, which I do before transmit).