I’d say the “religious” right isn’t the “core” of anyone.
Fiscal Conservatives are a Core
Judicial Conservatives are a Core
Economic Conservatives are a Core (different in that they want to see the economy grow, not government spending decrease)
Foreign Relation Conservatives are a Core (immigration, alliances, etc)
Domestic Security Conservatives are a Core (aka, war on terrorism, military spending, military development and research, etc)
Governmental Conservatives are a Core (as in they want smaller government with less power and states with more power)
Of those 6 pillars of Conservatism all have religious constituents as well as secular constituents. However, I do not see religion as a core of the republican party. Partially because that would imply that Christians want to impose a national religion and that is patently anti-conservative as it violates just about every one of the core constituencies.
The “religious Reich” are a bunch of religious people who want less restrictions on practicing their faith, less government influence and more state support.