The problem for the British monarch is that they are the supreme governor of the C of E. It comes with the job. A monarch in communion with the Church of Rome is one that acknowledges the Pope as both the Vicar of Christ and whose authority is final. So the monarch would be expressly subservient to the Pope. This is clearly an impossible to breach conflict with the position and oath upholding the C of E church.
The minor ins and outs of the specifics of the ecclesiastical rites and teachings are much less important. Although the modern C of E is moving ever further away.
Conversely the Roman Catholic attitude to the Anglican church is much more conciliatory. Alone in the Protestant and non-conformist churches, members of the Anglican church can convert to Catholicism almost without effort. To the point that ordination of Anglican ministers is recognised and they only require some basic retraining. How do you find a Catholic minister that is married with children? Find an ex-Anglican minister that converted.
Part of this is that the Pauline lineage of bishops is unbroken within the Anglican Church. So the authority of Anglican bishops to both create new bishops and ordain clergy is recognised.
Henry VIII died believing he was a devout Catholic. Just one that had a problem with papal authority. Fundamentally the entire point is to not allow that authority back.
It is an unusual situation, but any country that has a state church will run into similar problems. At least we are past God kings. Just kings that rule by the authority of God.