I'm not a Catholic, but in my opinion it has nothing at all to do with that. Christ laid down specific procedures to go through with a wayward brother. If all else fails the person is to be treated "like a tax collector." God wants members of his church to behave in a certain way while on earth. I have no idea if Congressman Obey truly belives or not - and that may be his biggest problem.
It's not blackmail; no one is forced to belong to the Catholic Church.
Catholic politicians who openly support and vote for abortion give other Catholics the impression that abortion is just dandy.
Abortion is a violation of the moral law and is condemned by the Catholic Church. So the Church, in order to prevent the scandal of a Catholic politician contravening a Church law, publicly, is beginning to say that Catholic politicians are free to vote for abortion, but are no longer free to call themselves Catholics.
Life's full of all kinds of choices, some easy some painful. This is a painful one, but politicians must make it.
It is not blackmail. The Catholic church has a responsibility. To turn and look away from one who claims to be a member then have that same person proceed to go against the church's moral teachings is akin to being an accessory to a crime.
Further, if it admits such a person, it weakens its teaching by seeming to say "we didn't really mean it." So it harms other people who need to see that it means what it says.
I know people, clergyfolk, who have left particular denominations over matters of conscience, sometimes incurring financial loss and the enmity of family and friends. It wasn't easy for them, but they did it.
They didn't think the church they left was wrong for sticking to its guns generally. They thought it picked the wrong guns to stick to.
But your argument seems to come down to, "If the church is consistent and sticks to its guns, that's blackmail." Is it blackmail for the Church to have an opinion about right and wrong, or to act on its opinion, or what? If the Church excommunicated someone who advocated slavery or persecuting Jews, would that be blackmail?
A group assembling, as the constitution says it has a right to do, says that they have certain requirements for members of that group. That's not blackmail.
If I marry THIS woman and am faithful to her, I am not thereby blackmailing all hordes of women clamoring to share my bed. If I buy a GM truck, I am not blackmailing Ford. If I start or join a golf club, I am not blackmailing bridge players. If I think abortion is very wrong, I am not blackmailing abortionists any more than they are blackmailing me when they do their filthy work. It is not blackmail when I reach a moral conclusion and act on it -- seems to me.
If my parents and siblings and wife and child turned out to be in favor of persecuting Jews, it would be hard for me to renounce them, but I would not be a victim of blackmail, except in a figurative and imprecise sense. If the politician's support for abortion costs him the approval of his denomination, then if he is a man of principle, he should leave the denomination. Yes it may be hard. But what good are principles if they are renounced when they prove difficult to uphold?
What am I missing in your point of view?
The only problem facing the Bishop of Milwaukee is that the other bishops of the American church are not doing the same thing.
BTW are you aware that in 1963 Louisiana Democratic leader Leander Perez was excommunciated by the Archbishop of New Orleans for publicly decrying the diocese's decision to integrate its schools. That action was widely praised by the Liberal elites in the U.S., and there was no ex cathedra teaching to compel the bishop to integrate the schools. It was a matter of the bishop's own sense of justice. Here the Bishop of Milwaukee has the guidance of two Popes, Paul VI amd John PaulII, speaking ex cathedra to guide his direction. Blackmail indeed. Disagree with the Church's view on abortion, but don't publicly facilitate the practice of abortion and expect to remain a Catholic in the eyes of the Church.