Congratulations on your growing family!
My initial reaction is this: Your first loyalty must be to your son and you must parent him as you see fit. You have been his primary parent. It is unfair to your son and you to have a "new' person come into an established relationship (that you are happy with and think is working well) and change things.
If your fiance starts making new rules, it also will only disrupt his relationship with your son. How you discipline your son is, in some ways, none of his business. He only needs to enforce your rules, and it should be clear to your son that you are setting the standards for his behavior, and that your fiance will only follow through with your expectations.
I also assume that some of this is coming from your fiance being a new parent. Frankly, he may just not "get it." You have a history since infancy and know your son better than anyone. You have a well-established working relationship with your son. Your fiance is looking at it as an outsider, even if he has been in your life for a few years.
All that being said, you have to find ways to look at things objectively and see if there is some truth to your fiance's observations. He certainly has a right to establish some expectations for dicipline in the household, as the rules you set now are going to soon influence your/his daughter too. And the teen years are going to be new and require new approaches anyway.
I think you and your fiance need to take a very rational approach to this. Have a parent's meeting and set some specific rules that address your fiance's concerns and still are acceptable to you. And then YOU have to follow-through. As I said, the boyfriend/stepfather should not get to be a disciplinarian with your son. He just needs to back you up and support YOUR rules. (And of course, if your son's father is in the picture, you need to be supportive of his views too.) You have to work with your fiance, independent of your son, to set consistent household rules and expectations. But your son needs to see the rules coming from you.
I'm not suggesting you compromise your beliefs about your son or about the way you parent. But you need ot make room for your fiance's approach. I just think it is super important that you parent your son and not your fiance.
Lastly, you mention a "hot temper." You also need to set some expectations with your fiance. he can get ticked off at you for the way you handle things, but he cannot undermine you just because he disagrees with you.
I also hear "hot temper" and worry that he might get physical wiht your son, which I hope is not the case, but if there is any issue related to that, you need to reconsider your plans.
Good luck!