Foodstamps/Medicaid/Custody issue in FL

I am the non custodial parent and make too much for foodstamps/medicaid and so does Childs father. However childs father wants me to let his girlfriend (they also share a kid together) claim my child on her Medicaid/foodstamps because she doesn’t work and will help out my child more in the long run. My question is I’m fairly sure this is going to end up in a custody battle very soon due to them not letting me see my child. Will letting her claim my child for benefits have any correlation to giving up my custody or lead to it, or make me look bad in court? I don’t want to be unknowingly signing any kind of rights away or benefit them in court