” But I still think accepting Christian Syrians make sense.”
They can lie and say they’re Christians.
Sure they can. But I'm not suggesting we just take their word for it.
As I said, the Christian communities over there are much more insular. The local priests, bishops, etc. are all known. So, you can start from there and get information from priests verifying who are members of their churches, for how long, etc. Once you have some verified members of those congregations, you can then use those folks as a resource if someone else claims to be a member of the congregation. There are other people who are well-known, long-standing members of the local Christian communities.
In short, we wouldn't be admitting self-professed "Christians". We'd be admitting verified members of specific Christian congregations. I think that's entirely doable.
It would really be no different from admitting Jewish refugees from Europe. A particular candidate says they're from a certain place, but when you check with other Jews from that location, they've never heard of the guy or otherwise cannot verify that he's Jewish. That's a "no".
I personally have a very difficult time with knowing that there are Christians being heavily persecuted for their religion, and not giving any of them a way out.