If she asks for court-ordered child support than the court will perform this test for a nominal fee between $25-100 depending on the state.
the public defender for his current criminal case won't pursue this because this would be a civil case. but he may be able to give him some info for a civil or pro-bono family atty.
there may be a way for him to initiate a civial case bu tin my experience it's usually when she's asking for child support.
if she's willing there are private firms but all 3 (pa, ma, & child) parties have to be willing and they can cost up to 600.