I am not sure, but I think, I hope, you missed my point.
Do you believe that The Church should be actively involved in politics as opposed to teaching not only its communicants but also society as a whole Christian principles and considerations for the proper functioning of society? If so, and I think that the Latin Church has slipped into exactly that here in America (and into a certain unseemly political stridency too), we have one more difference to deal with before any workable reunion.
Yes. The Bill of Rights puts limitation of government to not become involved in religion, but there is to reciprocal requirement. If corporations, labor unions and a host of other interests and advocacy groups can be directly involved in the political process restricting direct involvement by the Roman Catholic Church, or any other church for that matter, short changes Catholics and other religious persons and robs the process of the moral clarity and direction the Church can bring.