Is there a way I can choose for him not to be put on child support

Yes.

You get married.

Otherwise, he SHOULD be put on child support. The taxpayers don't want to pay to raise your child when the father should be paying child support.

Of course, if you get married, then you may not qualify for TANF because it will raise your family income level.

No you can not choose when your asking for getting help from the government. Why wouldn't he want to pay to provide for his child?