I'll go out on a leg and ask - why not brute force and use signal jammer. Or in SCIF, jam all incoming signals, except for the ones they want to whitelist. If there is wifi, whitelist devices that can be connected to it.
On an side note, like other federal laws, the DoD is subject to endangered species laws, and actually take a lot of care to make sure they do not endanger them, which is made fun of here:
https://terminallance.com/2017/01/17/terminal-lance-453-ceas...
Back when this controversy was first stirred up it seemed unlikely that anything sensitive was really exposed. Other comments in this thread indicate that the military is perfectly capable of restricting the use of personal electronic devices when they care about it. This seems like a case of the public demanding "something must be done" and the military responding with "okay, we did something".
Military bases already have rules covering use of electronics like laptops, cameras, and phones, so adding smart watches to that list isn't very surprising.
Jamming is done at the receiver, not the transmitter. The fact that the power is so low makes it far easier to jam.