You are correct, you can qualify if your meet the qualifications outlined in the instructions for form 2441 even though you are filing married-separate.
I got it to work by indicating that the taxpayer was married but filing separately and did not live with the spouse any time between July 1, 2019 and December 31, 2019. That is one of the stipulations in the regulations I believe.
I did indicate that the other parent was claiming the child as a dependent and I still got the child tax credit.
So, I think the issue is that you are living together, that is what is disallowing the credit.
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