FOBs are just a Bluetooth frequency receiver/transmitter). I could see how they could drive the bike nuts, but not each other.
I recommend creating a Faraday box for your wife's spare. An Altoid tin covered in a layer of tin foil is thin, easy to carry, and should do the trick.
— Never cross a "bridge", without checkin' out the view! - me
— Life is too short to drive an ugly car. -me