The oath you mentioned is as follows: "I swear by almighty God that I will be faithful, and bear true allegiance to his Majesty King Charles III, his heirs and successors, and that I will as in duty bound, honestly and faithfully defend his Majesty, his heirs and successors in person, crown and dignity, against all enemies, and will observe and obey all orders of His Majesty, his heirs and successors and the generals and officers set over me."
Heirs is in plural, meaning that unless you have renounced your oath, you are duty-bound to defend and obey the orders of Prince Harry, among others - no doubt you will do the right thing and defend the prince from these threats you are reporting, even if it is just a bunch of old farts on message boards and not active-duty military.