In setting, the Red Hooves would absolutely be inquisitors, among the church's duties is the suppression of pagan faiths so they do indeed have divisions that do exactly what you describe.

It's not impossible to find out about a being such as Santa however! The Vault almost certainly has some books that include those legends in their vaults, and some nobility have access to the old books since they're able to get their hands on more contraband if they so desire. The knowledge would EXIST, it wouldn't be necessarily easy to come by.