There are a number of problems I have with this bill:
1) It is unimaginably anti-Catholic. It does not seek to regulate any other Christian or non-Christian church. It's only about the Catholics. This makes sense considering that Connecticut has quite a substantial amount of Catholics (practicing and non).
2) It is obviously part of a larger liberal agenda. This bill is being introduced precisely because of the Catholic Church's stance on major liberal issues - abortion, gay marriage, embryonic stem cell research, etc. If the liberal government of Connecticut can infiltrate the Church - their number one enemy on social issues - they can win the culture war.
3) If this succeeds in Connecticut, it can succeed elsewhere. It is very possible that if Connecticut falls, New York, Vermont, and California will not be far behind.
4) It is a clear violation of the First Amendment, which specifically states that no Church will be run by the State. Although this Amendment has often been interpreted in reverse - that is, that the Church should have no say in State affairs - that is absolutely not what it is saying. The Church can and MUST have a say in Church affairs. The Founding Fathers were Christians that envisioned a Christian nation, far from the persecution of Europe. The First Amendment says that the State will have no say in the Church - and that is exactly what this bill is trying to do.
Which is why Connecticut's name as the "Constitution State" is really ironic at this juncture.