That said, my clock stopped syncing for some reason, so ...
As I understand it, the US switches at the same day across the country (for the states that do have it), so that should theoretically be possible.
https://www.nist.gov/pml/time-and-frequency-division/time-di...
So instead of a switch to “add an hour”, there could be one that says “add an hour if the signal says DST is currently in effect”.
[1] https://en.wikipedia.org/wiki/WWVB#Amplitude-modulated_time_...