After some kernel hacking, I was able to get the board to work with my Raspberry Pi!
I made some measurements using Room EQ Wizard with the board looping the outputs back to the inputs. REW only seemed to work in 16 bit mode, so the THD+N measurements are worse than what I measured with the Teensy. Looking at the CS4272 datasheet, the dynamic range is ~16dB worse using 16 bits, and the THD+N that I measured with REW in 16 bit mode is ~-84dB, so a 24 bit measurement should be right at what I measured with the Teensy (98-100dB).
The updated kernel code is in my fork of RaspberryPi's linux repo: here