I was actually in a similar situation OP. Had a baby with my long term partner (unplanned). Kept bringing up the subject of marriage and there was always a response of yes eventually, not now etc. I realised that it was never going to be a good time for him and basically gave him an ultimatum.
I tried not to let the emotions get in the way just said fairly calmly that as I didn’t want to continue to stay together unmarried that I would be moving out, the alternative was that we make concrete plans to get married.
Although he wasn’t thrilled with being given those two options, he did choose to marry me and we have now set a date to get married.
I know it’s not the ideal, I would have liked a romantic proposal but the principle for me was is he willing to compromise and do something he doesn’t want to do for my sake or not? I felt that would be a good indicator of how our future would look going forwards, more so than a feeling based proposal.
I think with some people it’s the concept of marriage maybe the finality of it that puts them off and is something they may never choose willingly, without an external reason to!
Whether he would have made the same decision if the baby wasn’t involved I don’t know, but I did reassure him that I would move locally and he would still have regular access etc, I didn’t want to use the baby as a tool to force his hand.
I did my best to mentally prepare for if he still said no to marriage, I had started making plans to move out and figured out how I would cope financially and with childcare. I think it’s important to do this, if an ultimatum is given you have to be prepared for either choice.
I don’t know if this is the right choice for you in your circumstances but I can only share my experience. It was getting hard to not be resentful towards him in the face of all the uncertainty and I don’t think I could have stayed much longer either way.