What about just covering the Nest with a towel or something for Shabbos? Would this fix the issue, and what would the affect be of it's ability to cool/heat?
Seriously?
A) Super ugly and it would fall off
B) Easier to just change the setting in the menu
C) It would make the sensor think that it's super hot in the house. In the summer - it would run your AC the entire Shabbos. In the winter -your heat would never turn on.
I was in a yeshiva that had the thermostats locked so the bochurim couldn't change them. We'd hang a hat over it so it would think it was really hot and turn on the AC

Any thermostat would do that if you covered it with a towel.