Should I give my son his father's last name?

So my bf and I are expecting a son in Aug. He has 2 children already and the state garnishes his income for them. In the state of Illinois, if a mother is receiving cash assistance from the government the state will seek child support from the non custodial parent. With nearly half his monthly income gone, he has decided not to sign my son's birth certificate and I am beyond upset about this. So I have decided that I will give my son my last name. I feel like, how can you want your son to carry your name if you don't even want it to be known that you are his father legally? I feel like my son is getting the short end of the stick because he wants to dodge a potential child support order, though I'm not planning on seeking child support unless he does not support his son. He doesn't realize that his name doesn't need to be on the birth certificate in order for me to get child support. So, should I give in and give my son his dad's last name or keep it in my family and name him after me?

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