Any idea how much the built-in amplifiers might be influencing distortion performance
Probably quite negligible in the frequency range in which we see most of the 308P's distortion. Even cheap-as-chips class D amps don't tend to produce more than 0.1% or at worst 0.2% HD except at very high frequencies.
Graph from datasheet of the all in one DSP/DAC/amp chip the 3 series uses.