While a bishop could arbitrarily excommunicate them, unless they've committed am obvious sin the Church is not going to invite a PR disaster by kicking out super rich members. This group is free to police only gay priests if that's their prerogative, and it'd still be an independent group using their own money.
I'm not talking about excommunication. Also, that penalty doesn't make an organization "non Catholic".
I'm talking about the Church's right to revoke the term "Catholic" which is not infrequently invoked, for example against RealCatholicTV and the National Catholic Reporter. It's not excommunication, just a representation and naming issue.
And I think the term you want is "financially autonomous" because no Catholic individual or group is "independent" unless they are schismatic.
The Catholic Church, of course, cannot trademark or copyright Catholicism or its doctrine, and so in most locales she has no secular legal standing to restrict the use of the word or term "Catholic".
However, the bishop can and will have a say in which Catholic groups call themselves "Catholic" and represent themselves as Catholic groups within the Catholic Church, and so they have ecclesiastical authority to revoke that right of representation. Not a few groups have abused the honor and privilege of the Catholic name, and it's been revoked.
Now also, whether or not a group chooses to cease calling itself "Catholic" is up to the group entirely, and whether they choose to be obedient. The National Catholic Register notoriously brushed off their bishop's concerns and continues to use the name. Others simply rename to something that doesn't contain that word.
It's also the case that the bishop gets to say who can reserve the Blessed Sacrament, and have it in a chapel where Mass is said. For example, St. Joseph Hospital and Medical Center in Phoenix had those privileges revoked, along with their Catholic identity, when they performed an abortion. That identity was later restored by the same bishop who revoked it, so it's all good.
