It would be cosmetically and structurally messy, but could you put an emergency variable incidence adjust or on the canard as a means to break a deep stall?
COULD you? Sure - if Starship could have a swinging canard, you
COULD do anything. As you say, a structural and control system nightmare, though.
Far easier to just determine the correct CG range for the airplane after a thorough design process that takes deep stall analysis into account, and then keep your **** CG in the right range

. It's not magic, or difficult. ALL the deep stalls that have occurred in Long-EZ's, Velocitys and COZYs can be attributed to incorrect CG location, incorrect CG calculation, unaccounted for fuel shift that moved the CG aft, leaving off required aerodynamic parts of the plane (Vortilons on the Leading Edge of the main wing) and/or incorrect canard incidence.
Build the plane right, fly it in the approved CG range, and no deep stall will occur. In Phase I, we're supposed to test our planes to the full extent of the operating envelope - I've stalled my plane in 60 degree banked turns at the aft CG limit with no safety issues (but this is a proven plane).
For a new design, a movable weight, as Cameron indicated Nat used in the COZY MKIV deep stall testing that led to shortening the canard span, is far less difficult and intrusive than a variable incidence (feathering, essentially) canard would be. Move a couple hundred lb. 6 ft. forward and your CG will move substantially to assist in deep stall recovery.