The Pope can teach infallibly but that does not mean that all his political decisions are infallible. Not at all. He, as a political leader, has all the same worldly forces bearing upon him as any other.
Do not forget that it was a pope who abandoned the Jesuit Order and then even shamefully suppressed it.
So his refusal to recognise a legitimate king is by no means decisive.
However, as a matter of fact, the Pope did recognise the Bonnie Prince as rightful king until such time as it was no longer politically expedient. Europe was in and out of war and the Pope had to be careful to maintain what friends he could.
Loyal Catholics knew what the true position was i.e. that the real king was Charles and not the Hanoverian usurpers.
The Pope was careful not to make any act of deposition or to call the claim of the Sutarts unlawful. He merely recognised a status quo and that for reasons of state - no more.
Thus there is no sense in which the Pope had censured any Catholics who were loyal to the true kings.
Indeed, the Canova momument to the Stuarts in St Peter's Basilica was permitted with papal approval.
When George I came to the throne - and that by the disgrace of Parliament and not the Grace of God - there were no less than 57 better claimants.
They were all disqualified for no other reason than that they were Catholics!
That, I think, says all you need to know.
He then gives a thoughtful clarification as to how Jacobites may support the current English monarch. As I noted in a comment, I believe the heart of his defense is aptly summed up thus: "Peace and Justice are higher goals than legitimacy."