Cool project! In principle I bet you could get a Bela-like arrangement going with your board, though it would involve a whole new driver compared to ALSA.
Bela uses both the McASP and the PRU. The McASP actually streams the data to the codec, but the PRU handles passing data to and from it, as a sort of sophisticated DMA controller. We use the PRU here because it can also do SPI transactions with the other ADC and DAC and it can sample the GPIOs, all aligned on a sample-by-sample basis with the audio clock. The PRU then puts the results in a memory buffer for the CPU to work with.
To take any advantage of that, you need a Xenomai kernel, because this lets you write audio code that can interrupt the entire rest of the OS. If you don't have this, then you're stuck with the timing uncertainty that regular Linux provides, meaning you need larger buffer sizes (= longer latency). It also means that audio performance may depend on system load, which is not true of Xenomai (rather the reverse: rest-of-system performance depends on audio load).
The PRU and Xenomai combination together represents an alternative to ALSA. If you use ALSA, you're stuck with normal Linux kernel priority and all of the above-mentioned performance limitations that go along with it.
If you wanted to port Bela to your hardware, you would need the following:
Possibly changes in the BB-BONE-BAREAUDI-02.dtbo overlay to initialise the McASP pins as you need them for your cape.
Modified PRU code which works specifically with your codec. The initialisation routines would change the most, and there might be some changes in the main data transfer loop. Presumably your board won't have the same SPI ADC and DAC as the Bela cape, so that code could be removed.
Changes in RTAudio.cpp and I2c_Codec.cpp to initialise your codec, which would be different from the TLV320AIC3104 codec we use.
A few parameter changes in the Bela core code to reflect a different number of channels by default and maybe different sample rates.
Your project is already open source, but just a note here for anyone else thinking of porting Bela to their hardware: Bela is LGPL, so changes to the core environment also need to be released open source if you're distributing it. But we'd love to see the Bela environment running on various audio boards!