Apology, act of contrition
I was running late. My apologies for passing bad info. I had coffee in my hand, food on the stove, a contractor running late, and an appointment I was rushing out to.
We know you have a choice in Dopers. We welcome the opportunity to earn back your trust, and we thank you for flying DavidNRockies 
I was thinking exactly what Joey_P said about airflow. I just didn’t get it out.
There are no end of locking thermostat covers available:
They’re just vented on the L&R sides and the top and bottom. You could do something similar with a foam cover by Xacto knifing slots in the perimeter.
Even the plastic type in the picture could dampen the sound enough to make you happy. The price to find out isn’t very high.
The other question is whether the existing t’stat is compatible with a remote temperature sensor*. You could foam box the t’stat in (w/Velcro or similar), and remote mount a temp sensor (which should operate silently) so that the operation of the system isn’t affected.
Or you could just get a new thermostat 
*This kind of thing:
https://www.amazon.com/Honeywell-C7189U1005-Indoor-Temperature-Thermostats/dp/B003UJYZDG