Or Faraday cages for entire rooms in my house!

Faraday cages around the property line. In LA some people already live in houses hidden behind 25 foot tall hedges. They need only a tin foil roof.

You could always try disconnecting the antenna that goes to the wireless module instead.

Disconnect may not be enough, best is to wire a resistor in place of the antenna, of the same resistance of the antenna, which will shed the attempt to use the antenna as heat.

(I disconnected the antenna on my OnStar, yet found it still connecting to cell towers successfully from time to time. The resister solved that.)