Like water spilled on the ground, which cannot be recovered, so we must die. But God does not take away life; instead, he devises ways so that a banished person may not remain estranged from him. 2 Sam 14:14
I don't know if someone has already proposed this. The problem of HH is essentially a problem of very thin deck. The main strategy to obtain a very thin deck, is using the Governor card and putting cards in reserve. Then a possible solution would be:
The main strategy for the HH thin deck is not the Governor. It is the reserve.
2: A player may not place more than two cards in reserve than your opponent has. Example: If your opponent has no cards in reserve you can put two cards. More if your opponent take the cards of the reserve is not required to taken as well. But can't put more cards.
British typically only place 2 cards in reserve to execute the HH, so this would serve to have no effect.