I think it's on him, to be honest. I also think, while responsible pet ownership is important, people come before animals. We don't always know how our lives will go. I'd at least try to responsibly re-home pets before I'd delay progress in a serious relationship.
That said, it's a shame he took in two cats after the two of you had already been together for a year. By then, it seems he'd have had an idea of where this would likely be heading and that the cats wouldn't merge well with the situation you already had- (allergies, a large dog, and free rent in a small place). Therefore, I'd say that if there was irresponsibility here, it was his, so he should handle it.
I also agree that moving the cats into your current home doesn't seem like a good idea. You're allergic to them and space is limited. Also, I've heard of more than one incident where a large dog actually killed a cat. I don't know how long it would take for your dog to recognize the cats as "family" but before that, the cats could be at risk, if they weren't able to be kept securely away from the dog. (Also, this is assuming that the owner of the free rental home has given permission for another person to live there, let alone two more pets).
If he can find a good home for the cats, that might be the best way to proceed here. If he personally knows someone who could possibly take them, it might help if he sweetens the deal by offering them some money to help with the costs. Perhaps a set amount per month, for the first year.
Or consider a third option. I don't quite get moving in together to save for a wedding anyway. It seems like "putting the cart before the horse," especially when there's a child involved. Living together isn't nearly the commitment marriage is. Men will more easily move in than marry. So, especially if your child is younger than mid-teens, it's a less stable situation for them than marriage would be.
Anyway, since this move is "to save up for the wedding," it sounds like it might be temporary, anyway. If the two of you plan to get a place of your own after you're married, why not just get married first. You could opt for a small, inexpensive wedding and use the money saved for a larger place that accommodates both of your current lives better, where there's a sufficient area that the cats could be contained in. We all value different things but I just see big, expensive weddings as more for first timers, without kids. Otherwise, it seems (to me) not very practical to blow that much on an elaborate party, when you have more grown-up lives, with more important needs. Just my thoughts, for what they're worth. Best wishes.