If they are not entirely distinct, they aren't distinct at all. The Church isn't a Frankenstein monster; it can't have heretical appendages hanging out of it. Either the Conciliar Church is a schismatic church entirely separate from the Catholic Church, or it is not a church at all but just a term describing group within the Church.
This is basic ontology. Even if you had a mad scientist somehow genetically blend a dog and a cat and make a dogcat, that dogcat is substantially a new entity. It may have some accidents of cat and some accidents of dog, but it is essentially something new.
No, the only way to view two churches as one thing is to view the one thing from two different formal aspects, i.e. to make a formal-material type of distinction, where it remains materially the Catholic Church but formally NOT the Catholic Church.
So, as you point out, it's either 1) the Catholic Church or 2) the Conciliar Church or 3) a brand new Frankenchurch, as you put it ... just like the dogcat is not both a dog and a cat at the same time, but a new thing called dogcat ... which has some accidents of dog and some accidents of cat. So too Frankenchurch would be a new thing altogether, neither the Catholic Church NOR the Conciliar Church, but something entirely new.
This ontology is deal with the first few weeks of any introduction to Thomistic-Aristotelian philosophy.