I'm just on page 1, but OP I don't think he does want a second. Look at his actions, not his words. Any man in his mid-forties with a partner approaching their forties who has already been through having one child knows the score, he knows it's quite possibly now or never. Like you say, he's risking your health with these silly delays. So I genuinely think it's that he doesn't want it and he's delaying telling you until you're too old to be able to split and meet someone else and have the chance of a second, sadly.
I had a kinda similar thing, only we planned two, had one, then when 6-12m after our kid had passed I mentioned saving bits for the next one, and DH said he had realised he was one and done. It took several years honestly of trying to work through it, oscillating between him trying to convince himself to want a second and me trying to convince myself to being happy one and done, weighing up what life would be like if we split or stayed together, at times DH said look, he can't have another (there were lots of valid reasons) and he gives me his blessing if I need to leave cos I can't be happy with him anymore, at times DH said he would try but only if I knew he would be hoping it didn't work which I obviously couldn't do. I wanted him to want another and could never have gone through with trying with a father that didn't want the baby. All the while seeing close friends get pregnant with seconds, thirds, and just grieving and aching for the potential baby while hating myself for not being present and fully joyful and happy with the amazing child I already was lucky to have.
We had marriage counselling and it was helpful as it enabled us to both address the issue instead of silently processing, seething, suppressing, it forced us to talk through our feelings and thoughts and felt like we were tackling the problem together for a change rather than as opposite sides. It did help and was well worth the money.
Eventually I had to close the decision myself, and tell myself that we're one and done for good, so I could grieve properly instead of having that glimmer of hope. And ultimately after time I started to genuinely appreciate being one and done and come round to DH's rationale. After our child started school I began to see so many other OAD families and feel less like an anomaly depriving my kid of a sibling. I'm at the point now where if DH said oh I've changed my mind, let's have another, I think I would decline, for my own reasons this time, not his (I could just be kidding myself, I guess I wouldn't know for sure unless that happened but as far as I know, this is how I feel).
I'm a year younger than you but with health issues that means I'm likely infertile anyway and was lucky to have one, so that's easier too as it's another factor closing the door.
But I had to decide for myself... do I want to be with this husband, with our child, or split, and possibly not meet anyone else in time, and have a blended family? No, I didn't want that at all. I wanted another with this husband. And if I couldn't have that, then I want us together regardless. I've seen other families where the dad does very little and he's happy to go along with whatever the wife wants, however many kids she prefers, because it doesn't materially impact his life in any real way. He doesn't change his hobbies, or work, or health, so what's another kid? Whereas for us, DH is a fantastic father that puts in at least his 50%, always has since day one. And if we had a second it would impact him as much as me in nearly every way. So I try tell myself it's better to have an amazing co-parent/father than one that is disengaged and doesn't take the question of another kid very seriously.
Ultimately OP you have choices, you can choose to accept this and stay, you can choose to leave, whether you end up with another or not. But that's for you to decide. Your boyfriend can't take a second away from you. It just won't be with him. If you don't want the relationship how it is now, with one child, you can leave. I highly recommend therapy together at least so you can get some clarity and figure out how to separate peacefully if it comes to that.
I'm sorry you're here, it's an enormously painful position to be in. I used to think I wish we could try and not succeed as at least we'd be on the same page grieving together. When one partner wants one and the other doesn't it's excruciating because it makes you into enemies on opposite sides. I felt quite embarrassed at times that my DH wouldn't impregnate me, like what's wrong with me?
But my DH was always open and honest and understanding, didn't waver or lead me on, it was clear to me we were both trying our best in a horrible situation, whereas it sounds like yours is being deliberately dishonest, leading you on while your fertility ticks away, and disregarding your right to make plans for your life by dangling it then coming up with reasons not to. I can truthfully say if my DH had acted like yours is I would have left. That's no relationship when the trust has gone.