Happened across my copy of Peter Berry's book on the Staggerwing; the deliberate gear up landings were at the instigation of Walter Beech to demonstrate the safety of such an arrival, and were flown by Bill Ong at the 1936 Los Angeles National Air Races and the 1937 Miami Air Show.
As Barit1 says, skids were fitted to prevent damage and also a brake to stop the prop horizontally.
BPF, thanks for the update, hope she is back in the air with a minimum of fuss and expense.