You named it yourself: Christian conservative
Or conservative Christian. Or Christian, if the religion is what you want to emphasize. Or Republican, or conservative, if you want to emphasize politics or philosophy.
When the words "right wing" get in there, certain connotations attach themselves, much more so than "on the right". Same with "left wing" vs. "on the left". I think it seems more like being "on" one side or the other implies you're across the street, and we can wave to each other, but wings imply that up-in-the-air-Junior-Birdman thing. If someone has a wing, is a wingnut far away?
It'll take a while to get this nomenclature straightened out, because the mouthiest on all sides have appropriated all the labels.