I don't see that your basic idea has been kicked apart, more that you have received suggestions on how you may make it work better.
I want to have just a few fertilized eggs to stick under a broody hen when needed, and to trade to other chicken folks for … stuff. Don’t need many, but the supply needs to be steady.
A steady supply is a challenge. Hens stop laying when they go broody or when they molt. As they get older they can slow down laying. There are techniques to get around some of that, though there will always be limits.
Pullets sometimes skip the molt their first fall and lay all through winter even without supplementing the lights.. Not all do that as I've experienced but many do. I rotate new pullets in every year and replace the older hens. I hatch my own replacements.
I don’t want to deal with a rooster that has control of the whole chicken yard
I'm not sure why this is a limitation. If we knew why maybe we cold discuss it more.
so I want to have a dedicated coop and run for a rooster and a few hens. That way I know the eggs coming from that coop should be fertile and I can use them as needed.
My question is: What is the fewest number of hens needed to make a rooster feel like the king of his coop without a lot of social chicken issues to deal with.
I will most likely be using Dominique's.
Why Dominiques? Have you identified a market for them? Some other reason? Knowing why or knowing your goals can be really useful when coming up with suggestions and possible solutions.
Regardless of the breed, some people look at a chart or even base it on their own experiences, usually with very few birds, and assume that every chicken of that breed is exactly the same. That is not even close to true. Individuals lay eggs, some many more than others of the same breed. Individuals go broody or don't go broody. You can get a lot of size difference within a breed. If a flock has been together for a few generations without bringing in outside birds you can get a lot of differences by flock. These is called strains of that breed. We all have our own goals.
As Mrs K said, "I have had the dominique - some people love them, I was not that impressed. To each his own." She is exactly right, she usually is. If her goals were different dominique may have suited her fine. If she had a different strain she may have liked them a lot more.
When selecting a breed (unless you know the flock they are coming from) breed averages may be the best you can do. But don't be surprised if the ones you get don't meet those averages that well. They may be above, they may be below.
I'm still not totally sure of your goals. If you want to enhance the value on the market of your hatching eggs a specific breed could be important. I was going to suggest you get a leghorn rooster and sell leghorn hatching eggs as a possibility. That simplifies it a bunch if you don't have to separate them. Another alternative would be to get Ameraucana hens and a rooster, that would give you blue hatching eggs to sell but Ameraurcana are not known to be a great laying bird. Not real big either for meat. As others have commented if the hens lay different colored eggs to the leghorns you don't have to separate them.
From some of your latest posts it sounds like you want the hatching eggs you sell or hatch yourself to be good for meat. Is that one of your goals? Cornish X are not good for keeping to get eggs, they grow so fast it's hard to keep them alive. They also grow really fast. You pretty much have to buy the chicks each time. Rangers can be a little easier but not a lot, you will still probably need to buy chicks. You are probably going to be a lot better off using a heritage type bird.
There are a lot of different ways you could go with this. I'm not sure I understand your goals well enough to make a good suggestion.