To keep it simple, not trying to complicate this
But essentially how can the word of a man in a small piece of territory in the middle of Rome be enforced in America? You mentioned barring him entry, taking assets etc.
Separation of Church and State can mean a lot of things in the US but it's especially meant for legal matters. In essence, Federal and State law has to defer to the governing laws of a Church when it comes to church matters. They cannot override them unless the public is involved (i.e sex abuse, etc.)
So if the canons of the Catholic Church say that all real and intellectual property, assets, and real estate held by the Church is owned by the Papal Seat, State and Federal law could not override that, and so the Vatican would not only have the legal right as the legal owners, but the courts would have to rule in its favor if a legal issue came out of it.
It wouldn't get that far though. There are probably four or five other Bishops in Texas along with the one in the OP, and they all answer to an Archbishop. The Archbishop, who has authority over this Bishop, would just step in on behalf of the Pope to have him physically barred/removed if it got to that point.
The separation of Church and State goes
deep. I grew up in the Episcopal Church which is the oldest church in the US. The Canons of the Episcopal Church are older than the US Constitution. This is not known to the general public but because 1) we have separation of Church and State and 2) because the Canons are legal documents older than the US Constitution, in instances where the Church may violate federal law, there is a quiet acceptance among the courts that the Canons take precedence over the US Constitution. And there have been one or two instances where I heard this actually happened (can't remember the circumstances, gonna look it up). It is the only document in the US, and I think there may be one more, that quietly has this privilege but it's not really spoken about and only comes up if there are legal matters within the Episcopal Church of a federal nature.