If they get 70% of their funding from government sources then they aren’t even a “charity” at all by any objective measure — let alone a religious organization.
Funding for the program not overall. Since CC and other private charities providing services are non-profits their tax returns should reflect that status and show they are not getting rich off of such funding.
Legitimate services for legitimate refugees are not in danger. Narrowing the definition of refugee means that more of those who are genuinely in need will be allowed in the U.S. which is how it should be. Most likely Trump has decided to stick with the correct definition under US law.