Should I give my baby his fathers last name

I’m debating on giving my baby’s his fathers last name, just because I don’t really feel like he deserves it and I don’t wanna end up giving him his last name for me just to regret it. First off in the beginning of my pregnancy he cheated on me with his baby mamma while being a broke ass bum not working driving my car me using my money for gas so he can see his “son” when really he was seeing his baby mamma tooo. I found out about it we tried to stay together but I couldn’t forgive him so we broke up but we stayed tExting eachother to maybe try and work things out, he’s been nothing but good to me after the fact I found out about him cheating, he will go to ultrasound appointments and if I needed help paying for one he’d pay half. I’m now 2 months and 10 days away from having my baby boy, and he just hasn’t really bought anything for him, he’s bought one box of diapers but the only reason was because that’s the only way I would hangout with him for a little bit and the other thing he bought was 140 dollars worth of baby stuff that’s it and that was because I was with him and pretty much made him do it. He just recently quit his job without having a back up. I just think he’s very stupid for that knowing he still needs to play his part and buy some more things for the baby! He’s waiting for the last minute on buying stuff and it really pissses me off, I just don’t know what to do about the situation I’m done reminding him he needs to buy some more things. My family thinks he’s a dead beat because he lived with us for 3 months without trying to get a job couldn’t even buy his 2 year old son Christmas presents because he didn’t have a job, but no worries my damily was nice enough to buy a couple things for him to give to his son! Still doesn’t give his baby mom money or buy anything for the baby because she doesn’t ask he doesn’t even pay child support. I’m just wanting him to prove that he’s not what my family thinks of him and so far he’s let me down every single time. He’s absolutely pissed that I’m debating on the last name because he thinks the baby should have it no matter what cus it’s his son. I’m really not letting him have a say in anything that has to do with the baby , I just feel like he doesn’t deserve a say in after he cheated on me