After careful consideration, I have decided to go for a single bias source, instead of two (left and right). The reference design for the AK5397 uses a single one, and I am worried that having two different voltage levels for left and right could create some unforeseen issues. This will make the PCB a bit less symmetrical than it would be with two levels, but better be safe than sorry.
Also, I have decided to get DVDDL and DVDDR (3.3 V) from AVDDL and AVDDR (5 V) respectively, instead of getting them from the +6V input, which should reduce the amount of power loss quite a bit. Finally, I have removed all capacitors from the audio input signals, as mentioned on this post.
I am now working on a 4 layer version of the PCB.
With a bit of luck, this should be the last major revision.