It's really tough to make sense of this because we (well at least I) don't have the foggiest idea about lay military religious orders as a class of historical Catholic entities. Given that V1 essentially gave the Bishop of Rome absolute jurisdiction over everyone and everything Catholic, I think you are right, Mike. Francis has the upper hand. 

For my money, the critical fact here is that Burke personally presumed to speak for Francis. NCR reports have never stated that, but the Tablet sure did:
"The respected German Knight was sacked on 6 December by the Knights’ Grand Master, Matthew Festing, in the presence of the Order’s patron and prominent conservative critic of Francis, Cardinal Burke. BOTH OF THEM had claimed that the dismissal was in “accordance with the wishes of the Holy See”. (emphasis added)

Festing claims neither the pope nor members of the order can criticize his action. That's more authority than Francis, given the V2 concept of collegiality, claims for the papacy.