I'd experiment with microphone placement. Please show us photos of how you have your mic mounted.

 

What do you think?

 

Microphones intended for use in a car usually have some kind of noise cancelling circuitry and if there's excessive noise it may garble or seem like the mic is cutting out.

Just as a test, try wrapping the outside of the mike with open cell foam or something to prevent it from getting vibration from the steering column or console. So many things may not seem loud to us, but might be vibrating enough that something touching it acting like a stethoscope can really be overwhelmed.

Another possibility is that there may be enough electronic noise in that area to affect the mic. As a test, you could try rerouting the mic to your center console or somewhere further from the dash just to see what happens.

 

You could also try posting your question at https://www.diymobileaudio.com/forums/general-car-audio-discussion.61/

 

Mine is wired such that the wire goes up the A-pillar, across/above the headliner, and pops out and clamps on directly above the rear view mirror. Works reasonably well for an inexpensive head unit/system....