The OP has said she's not leaving him which has changed my mind on this. If she felt so little for him she was prepared to chuck in the marriage and family for him I would actually say he was probably wise not to have another baby with someone so uncommitted to him. Because who knows what else they would decide to dump them for if they didn't get their own way. But the OP is fully committed which makes a big difference.
OP, have you actually communicated all this to him as you have written it here? I think that is the first step.
It seems to me like the problem is that he is not ready for a second baby right now but isn't ruling it out for the future. But because you're putting him under pressure he's giving you deadlines even though he can't know when he will feel ready. And then the deadline comes around and he still doesn't feel ready so he fudges it.
No matter what he has promised he still has every right to change his mind or want to delay until he is ready. I hate this MN idea that women have complete autonomy over whether to reproduce or not. But men have to comply with their partners wishes.
I think OP that you will have to accept it if his position is that he's not ready right now and he can't put a specific date on when he will be ready.
As an aside, I suspect that if you eased off the pressure it actually might make him more inclined to go ahead. As someone who's been through fertility treatment I can tell you that a lot of men find it very difficult to be reduced to feeling like a sperm donor or an incidental inconvenience necessary for their partners desire for another baby. Withholding sex would be a terrible idea as it would probably drive him further into being unwilling as he would feel like he was being pressured and blackmailed and forced into it rather than being an equal parent allowed to make decisions over it together rather than being pushed around and bullied into it.
Talk to him and tell him what you've told us here. And perhaps consider taking a year removing the pressure and enjoying the family that you do have before broaching the subject again. You said you are youngish so I really do think that you are way, way overreacting in terms of urgency. Just chill out a bit, that will probably make it happen much faster than pressure and demands.