But if they know they are threatened, would they risk putting people off who would not then vote for them (like me)?

Because there's always a tension between using the political power you have to get what you want, and subordinating what you want so you can get or keep political power. That tension is strongest for people who are in the base of the party (both parties). Those people engage in politics in order to try to get things enacted that are further to the left or right of the mainstream, but enacting them makes it harder for them to win political majorities.

Religious institutions and communities used to have enormous power in American society, much more than they do today. Because that power has waned so much, trying to influence government power is one of the few levers they have left to really try to effect changes in the country. It's hardly surprising they fight so hard for their goals in that arena, given how few other avenues they have to pursue those interests.