How can I mitigate SNR or THD for audio codecs?
Your signal chain performance is limited by the weakest link, and for audio that's generally the DAC and ADC. You need to be careful that the analog portions don't degrade the DAC/ADC, but that's eminently do-able.

The digital/analog conversion process produces artifacts that manifest themselves as spurious signals -- 'spurs'. It's spurs that contribute to your SNR (more serious parts will specify their SFDR -- spurious-free dynamic range). If you were dealing with pure tones, it would in theory be possible to map and ignore or mask spurs, but it's unrealistic in practice, and well-nigh impossible to do for an arbitrary audio waveform.

Bottom line is it's very hard to improve your DAC/ADC from the outside. (If it was easy, everyone would be doing it.) You can add dither to your waveform, which has the effect of reducing spurs, but it comes at a cost of raising the noise floor -- it's a tradeoff that needs to be checked on the bench. If your DAC supports higher sample rates, upsampling can provide better performance, but it really depends on the part -- analog performance vs frequency tends to peter out at some point.

The part you have chosen looks like it's aimed for consumer applications where nobody is going to look very closely at the technicals (it's only specced at 48 ksps). It just needs to sound okay. So if you want better performance, you need a different choice of parts.