I don’t think you’re asking too much to ask him to do ONE chore per week without you having to nag.
Here’s what I have done in your situation. It worked for me and my brothers and later with my husband as well, with varying levels of success (brothers=cared less what I think of them, thus did less for me but still more than nagging ever got them to do, husband=cares a lot what I think of him, thus caught on fast and now does pretty much everything I ask him to do).
Right now, you are doing essentially all of the chores in the house, and probably spending a lot of time/frustration/mental energy trying to nag your Fi into doing some work. Even with all your nagging, you say your FI still isn’t cooking, and does the dishes maybe once per week. Can we both agree that time you spend nagging is wasted, and that by nagging you are actually expending more time and effort than what it would take for you to just do that one chore your nagging induces him to do? Would it really be so terrible if you just did that one extra chore, and in return got the clean house and nag-less life you want?
Assuming you can get in this mindset (it takes some work), try this: stop nagging entirely and just ask him for help, ONCE only. If he helps, great. If he doesn’t, do it yourself. For example, you guys just finished dinner and you are hoping he’ll help with cleanup. Say, “That meatloaf was good, wasn’t it? I’m really tired. Would you mind doing the dishes tonight instead of me?” If he’s like every man I’ve ever known, he’ll say “Sure, no prob” and the dishes will still be in the sink the following morning. But give him a couple hours to follow through, just in case he surprises you. If he DOES do the dishes, thank him and say specifically, “It is so nice being able to rely on you for help in the house.” If he hasn’t done the dishes by the time you’re getting ready for bed or dessert or whatever, just go do them, but don’t say anything. If he notices later and plays the whole, “But I was GOING to do them!” game, just calmly say, “OK. But when I asked you for help, I was hoping you would do it relatively soon, not hours later. I wanted the dishes clean, so I just did it myself. But I wish I could rely on you for help sometimes.” Then (this is so important) drop it. No nagging, no guilt-tripping, no fighting. Oh, and if he hears you starting to clean and comes running over to take over from you, LET HIM. Just walk away and be like, “Thanks.”
The point is to try to break out of the nagging/defensive cycle that leaves both people feeling like the wronged party. This calmer, less-aggressive, nag-free approach has gotten me more help from the men in my life than any amount of nagging ever has. I think most men do truly care what their loved ones think of them, and they WANT to be seen as a kind, thoughtful guy who helps his woman. When you nag, though, the man doesn’t think at all of what YOU think of him, because his brain is too busy getting defensive against your nagging accusations (“You never help out! I do everything around here!” etc.)
The truth is, I’m not sure how feminist of an approach this is. I definitely think it would be a lot better if men just weren’t raised to be so entitled about having their housework done for them. Then again, some people are just naturally sloppy and care less about having a clean house, and need reminders to keep up with a cleaner partner’s comfort level.
I think this approach might not work if your fiance truly feels entitled to having his little woman take care of him as if he were a baby. For your sake, I hope this isn’t true.