Is this an SPI protocol?
I think I may have found the answer, after some more digging.
The RF chip seems to be PL1167.

Going through the register datasheet, I can see that register 0x23 is power management (first message that gets sent in my remote), 0x34 is FIFO pointer, 0x32 is FIFO data, 0x07 is TX enable, which coincides very well with what I'm seeing on the traces.

The inspiration came while looking at the MiLight reverse-engineering effort: https://hackaday.io/project/5888/logs and this led me to the Group discussing this chip https://groups.google.com/forum/#!msg/openhab/ayFUwuABpcE/ThiUv94bq9QJ