We are to give Caesar what is Caesar's, but must never give to Caesar what is God's. Once we choose where to draw the line, and decide we cannot let it be crossed, then we have to decide how we will respond.
One, we can leave a government's jurisdiction, as the Pilgrims and millions of our ancestors did. Twelve million came through Ellis Island. The dilemma here is; where would we go? Where would things be better than here?
Two, we have the right to plead our case in whatever proper channels are available to us. In the USA, nothing is wrong with seeking to exert influence by going to the courts. Paul appealed to Caesar. Hobby Lobby has justly taken to the Supreme Court its case to not have to provide aborticides.
Three, civil disobedience. If the first two options are not taken, we have no choice but to practice civil disobedience. I remember Baptist Pastor Martin Luther King Jr. going to jail. Our Baptist forebears suffered grievously in pre-Revolution years at the hand of government and established churches.
Lutheran Pastor Dietrich Bonhoeffer gave his life in refusing to cowtow to Hitler. His book "The Cost of Discipleship" is already considered by many to be a modern classic. He died by hanging in a Nazi concentration camp, only 23 days before the German surrender. He lived up to his own statement, "When Christ calls a man, He bids him come and die."