You aren't doing DS any favours by being soft, he's not going to stop loving his parents (NB plural) if you both keep him on the straight and narrow re: tidying this or remembering that and helping with chores.
OP I understand you want DS to have what you didn't - he has two loving parents, he's had a great head start already. Now before he heads off to secondary school you can ease up on the babying and smoothing his way.
The way it sounds at the moment it's like two very separate camps, you are in the middle. That's not really how a family works.
Incidentally are you aware or not that by placing yourself in the middle it's also a way of being the One Most Wanted. Both males in your life want your attention. It's nice to feel wanted but then the buck stops with you. You have defended H and still want to make this work.
The parenting issue might be a red herring but you can test H's commitment. Parenting as a couple means equal responsibility so trust H and give some control to him.
If things don't work out you'll know you tried. Whether or not there is an OW waiting in the wings H's job of being a father needn't stop the moment he packs his bags. Even if you separate as a couple, that bond (If there is one) can still exist and develop.
Personally I don't react well to being handed an ultimatum. I may not agree with his tactic but it's making you look at life with H, and how you are raising your DC. What are your thoughts OP?