Reflecting on this, it seems very hard and improbable. Not impossible, I think there might be a way, but it would require great courage on the part of the writer .
Let us recapitulate.
For the King to personally be a Roman Catholic , recognising the authority of the Pope as defined by the latter at the end of the 17C. Hard, contentious, but not impossible for clever and determined king. James nearly managed it and he wasn't very clever.
For the Anglican church to move to a position where it de facto (but not de jure) accepts and implements the teachings and dogma of the Roman church, on purely spiritual matters? The Real Presence, justification, Scripture Sola, purgatory and so on. Quite possible provided it be taken slowly. No forcible change is needed, just appoint bishops who progress in the desired direction. The changes are not actually that great, and laymen do not understand them anyway.
For the Anglican church to implement changes in ecclesiology to align to Roman practice. Bells and smells, vestments. An Anglican church that LOOKS just like a Roman one. Easy. Laud almost managed, the nineteenth century did.
An Anglican Church that accepts Roman canon law on matters like celibacy of clergy, services in the vernacular, communion in two kinds.Patchy. Some easy (eg celibacy) some harder. But in almost all cases the Roman church, post Vatican 2 has actually adopted Anglican practice . The big exception is celibacy which the CoE then would have been fairly willing to accept. Other matters which are different today, divorce, homosexuality , female priests, were not at difference then, both sides condemned them. So, overall easy enough.
But. For the English church and the English people to accept the temporal authority of the Bishop of Rome, on any basis that a late 17C Pope would be willing to offer? Impossible. I do not think any king could survive such an attempt, or if he did it would be at the expense of civil war that would make 1648 look like a minor squabble, and hangings and burnings such that *Bloody Charles would make Bloody Mary appear positively benign.
The problem is that it is not possible simply to rewind the Church of England position back to pre Henry VIII and reattach it to Rome. Since the time of Henry VIII the Church of Rome has changed and moved on itself. The reformation removed the dissident elements which pre reformation had controlled the ambitions of the Popes. They (the dissidents) became Protestant. The result was great hardening up of the authority of the Pope ( in their opinion, anyway). Even the practice of the pre reformation english church would not now be acceptable to any feasible Pope.
The King would have to go to Rome and petition the Pope, "Please can the Church of England be readmitted". The Pope would reply "Yes, but here are the terms". And they would be such that the King would not long be King. I cannot conceive of anyone who might be elected Pope of the period who would accept terms acceptable to the english.
And , therein, perhaps lies the answer. The problem is the Pope. Remove him from the discussion and all becomes easy. That sounds ASB, but perhaps it is not.
Now, King Louis rather liked a lot of what he saw in the English church. Bishops appointed by the King. He liked that. Control of Papal officials, royal authority over church lands, Louis liked quite a lot of those things. The result was the establishment in France of Gallicanism. Not quite the Anglican church of Henry VIII , but close.
Let us look at some of the principles of the Gallican church .
According to the initial Gallican theory, then, papal primacy was limited first by the temporal power of monarchs, which, by divine will, was inviolable. Secondly, it was limited by the authority of the general councils and the bishops, and lastly by the canons and customs of particular churches, which the pope was bound to take into account when he exercised his authority.
....
The Kings of France had the right to assemble councils in their dominions, and to make laws and regulations touching ecclesiastical matters.
The pope's legates could not be sent into France, or exercise their power within that kingdom, except at the king's request or with his consent.
Royal officers could not be excommunicated for any act performed in the discharge of their official duties.
The pope could not authorize the alienation of any landed estate of the Churches, or the diminishing of any foundations.
His Bulls and Letters might not be executed without the Pareatis of the king or his officers.
He could not issue dispensations to the prejudice of the laudable customs and statutes of the cathedral Churches.
It was lawful to appeal from him to a future council, or to have recourse to the "appeal as from an abuse" against acts of the ecclesiastical power.
Moreover in 1682 Louis had extended by Royal decree the droit regale to all French churches, which fave the King the right to appoint bishops.
All this would look very familiar to Anglicans. OTL the Pope stomped his feet and Louis by and large knuckled under.But, maybe, the influence of a friendly, cunning, Roman but not Papist King of England might encourage *Louis to stick stubbornly to his Gallican church.
And let us suppose that the Pope also pisses off the Emperor. Who already makes very similar claims, and remembers that the first Ecumenical Councils of the Church were summoned by the Emperors, not the Popes. Not hard to see that happening. The Cologne Controversy would be a good starting point.
Or perhaps the *Pope in a fit of hubris declares that if , as is expected, the royal line of Spain should fail, he, the Pope, would determine who should inherit the thrones of the Spains. Which would royally infuriate everyone . Popes had made claims as vainglorious within almost living memory.
So, the King of France (who OTL was around now pretty much at war with the Pope), the King of England and the Emperor decide jointly that the Pope has gotten too big for his red boots, and needs to be put in his place.
They convene a Council (let's avoid calling it an Ecumenical Council, that might be a bridge too far). They bring along their (royally appointed ) archbishops Amongst other things they resolve that the Church of England, so long regrettably sundered from the communion of the faithful should be readmitted. On terms which are fairly similar to those of Louis for the Gallican church (the Emperor might well decide that the Austrian and Hungarian churches could do with a bit of this also). The King of England will, over a defined period, implement some changes.The Roman Church will accept some local variations . The Pope will have no temporal power at all in England Scotland Ireland or France . Probably a "Me too" from the Emperor at this point. And if the Pope didn't like it, tough.
It would be essential though for England to retain the right of advowson. That would be a big big stumbling block for the Pope (since it totally destroys his temporal power in England) but it is absolutely essential.
That's a BIG change , a massive POD. Said it would require courage . But I can't think of a simpler way, other than pure hand-wavium.