The source code for this project is now available on GitHub: https://github.com/DigitalConfections/Receiver-Development-Platform
There is still plenty of software work to do, but it seems to be working nicely so far. The latest addition was drivers for the two real-time clocks: PCF2129 and DS3231. It is now possible to set the clock time using the Control Head's menu system. Just tap the knob to select the field you want to change (seconds, minutes, hours) and turn the knob to increment/decrement the field. After setting the time on the Control Head, you can view and then synchronize an attached Digital Interface's time to agree with the Control Head's time.
The receiver display screen shows the time elapsed since the headphone was plugged in. The photo below shows 1 minute and 11 seconds having elapsed since the unit was powered up. Elapsed time is derived by capturing the real-time clock's time at power up, and then subtracting that time from the current time reading. It rolls over after 24 hours, so you have to finish hunting before then.

I am a little disappointed with the PCF2129. Reading that device's registers seems to require reading different register addresses than what was written to - and different from what the data sheet says! A simple hack to the driver allows the correct time to be read, but that should not be necessary. By contrast, the DS3231 worked exactly as expected, using the exact same I2C read/write functions.
At this point, both clocks have been running for a little more than one day, with no discernable drift. But testing needs to be done over temperature, and over much longer time periods. But the clock backup batteries are in place, and the testing can begin.
I anticipate using interrupts (once per second) generated by the RTC for precise time keeping on the transmitter. But I still need to experiment with configuring the RTCs to trigger those interrupts.