Yes, the audio breakout is a crucial piece of information that was not shared.
That rules the Metra piece out for this situation.
The PVI modulators have a feature to reduce encode time, there for reducing lipsync issues. But it is not perfect and there can still be slight sync issues sometimes.

If perfectly synced audio is required, then a much more expensive solution HAS to be used. You could use JAP with the audio breakout, it allows you to time delay the audio so that it is perfectly in-sync. Or you could use an HDMI matrix with HDBaseT and audio breakout, like Zektor has.