I'm making an assumption. When you breed the hens with the roosters for hatching, you will isolate them by breed so you will get purebred chicks. Otherwise you will get mutts. Roosters don't care which breed a hen is. The different breeds are a manmade thing, not a natural chicken thing. You probably know this but I'll mention it anyway.
I'd suggest two roosters per breed. It varies by individual, but one full sized rooster can pretty much assure 10 to 12 hens are laying fertile eggs, some more and some less. With two roosters per 15 hens, you can be pretty sure all eggs are fertile. You don't need to feed more than two roosters per breed to assure fertility. With this, you will have a spare rooster per breed if something happens to one of them. You also mentioned genetic diversity. You get more genetic diversity with two roosters than one while I don't think you gain a whole lot of genetic diversity with three compared to two. The alpha probably will not let the two subordinants have that many hens.
Another thing to consider. When you decide to hatch the eggs, how many will you hatch at one time? Will you keep all the eggs all 15 hens of that breed are laying and hatch them, or will you select a smaller number? As you have seen, you can keep multiple roosters together. You also acknowledge that there can sometimes be problems with multiple roosters. I always try to advise to keep the minimum number of roosters consistent with your goals to minimize the chance of problems. If you are going to isolate the hens and roosters by breed when it is time to hatch the eggs, you don't necessarily have to get eggs from all 15 hens to hatch. You could put your best 8 to 10 hens with the best rooster and hatch those eggs. You lose some genetic diversity and a back-up, but maybe something to consider, especially if you don't want to feed the extra roosters.
Also, when you take a bunch of hens and two roosters from the general flock and isolate them, you are changing the pecking order of the flock, both the general flock plus the new flock you are creating. For the new flock it is possible the two roosters will have a dominance issue, especially if neither of the two were top alpha of the flock. I'm not saying they definitely will have issues, just that it is a possibility. And if you take roosters out of the flock for a time then put them back, they might have dominance issues when you mix them back in, or, especially if the rooster you took out was the dominant rooster, in the general flock itself when you take them out. Again, I'm not saying it is absolutely going to happen, just something to consider. Maybe keeping the roosters in a bachelor pen isolated from the females until you want to breed specific ones, then putting them back in the bachelor pen after you have your hatching eggs. This might make for a more peaceful flock once you start your breeeding program.
With certain assumptions and what I think your goals are, I still think two roosters per breed is your best bet, but there can be some management issues you want to consider. Playing it by ear and being prepared to act if necessary may be the best course.
Anyway, there are my thoughts. Good luck!!