He has no chance of taking custody from you, and is using that to scare you off of collecting child support. If he really wants to be a father to this child, he will have to jump through many hoops. First, he has to file a custody case (which is separate from a child support action). Then he will likely get supervised visits which he will have to do for quite some time. Whether or not he is ever able to progresses to unsupervised visits will depend on the nature of the offense that put him on the registry and lifetime probation (which indicates a serious offense). If he files custody papers, you need to at least consult with an attorney, and should hire one if you are able.
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Ms. Whitbeck's advise is very sound. I just want to add that you may not have to worry about the DNA test results if he is on the birth certificate of the child.