Some people are really missing the point here.
If the OP wanted to get married and that's all there was to it, of COURSE she could propose to him, but that's not the only issue here. She wants to feel like this is a mutual decision, that he actually wants to marry her as well.
he never tells me a reason other than not the right moment, or gets all coy and says he will do it “one day
Sorry OP but from this and snippets of your other posts I would be very surprised if you ever got married now. Like someone else said, he had you, a place to live with you, and a baby. He's basically ticked all boxes without having to marry you.
Some men do this. It's fucking rotten. They wait and wait, knowing that one day, their partner will get so desperate for a baby and worried about their biological clocks that they will have kids with them anyway. Even though fertility does decline in men from age 30, it's not the man who has to carry the baby so the woman has to consider that as well. They know this, and they know if that if they keep stringing her along, she'll probably be the first to crack.
Your situation was a bit different because it sounds like it was a mutual decision to try after 35, but you've still had a baby either because you assumed marriage would happen, or because you wanted a baby so much that being an unmarried mother was a price you were willing to pay on some level.
I wouldn't give the baby his surname at all, not even a double-barrelled one. It would be a fucking cheek if he did expect this. You are putting yourself in harm's way to have this baby. It could be your health that's affected. If you think you're going to be more financially vulnerable due to not being married and he's not enthused about addressing this, then no, the child is not getting his surname in my view.