there’s a lot to unpack here tbh and i apologise if i’ve misinterpreted or misjudged some of your meaning.
ultimately, like you say, this is not the child’s fault. it’s not their fault that they’re living in a episode of shameless, it’s not their fault that they’re the product of a “quick bang” (of which your partner was a willing participant) - your partner’s attitude towards this child, as you’ve described it, is incredibly poor tbh. he sounds just not bothered by the state their child is living in and is doing the absolute bare minimum. that’s his child! who is, from the sounds of it, in a fairly difficult and tumultuous home. the poor thing sounds like no one is really looking out for them.
that aside, i wonder what the true question is here in this post as while you’ve explained the situation very eloquently - you haven’t said what aspect exactly you want advice on. does his non-relationship with his child bother you? or is it the existence of the child themselves? when you say that you would feel unfair having a child with him while his other child is living an episode of shameless, is this because you wish that child wasn’t there, or that you’re secretly annoyed with your partner with not stepping up for this child? what is the dealbreaker that’s made you consider leaving him?