Actually the problems are not caused by what you mistakenly label "anti-theological organizations" but rather by theological organizations and individuals violating the constitution.
With such a non-defined subject as the intent of the constitutions reason for mentioning a seperation of church and state, how you see it in detail is no more valid than how I see it. So stop trying to make it sound like you or understand something that others don't. It is the non-descript way seperation of church and state is stated in the constitution that has allowed for the problems that anti-theological organizations have caused for those of us of faith.
Oh hell, every religion and religious organization in the land tries to influence politicians and legislation.
If the Catholic Churce wants to preach to Catholics to work together to change immigration laws, that is one thing. To go directly to Catholic politicians and tell them to change immigration laws is despicable. It may not be illegal, and it may be the right of the Catholic Church to do it, but it is still no less despicable than how anti-theological organizations have acted.