In terms of the behaviour problems, if I let this year's chicks breed next year and then again the following year, would there be less behaviour problems the second year since they would be more mature then? Or would they continue to have behavioural problems every year afterwards because I let them start breeding too young?
Next year will probably be better.
We would want to hatch maybe 30 or so La Bresse per year to raise for meat. With the other breeds, we would only want to hatch enough to replace older birds and generally keep the flock going.
Remember that about half of all chicks will be males. So you will end up butchering some males from the other breeds too, along with probably a few pullets (extras, or ones that are not as good as you were hoping), plus some older birds too.
So the Bresse will not be the only chickens you are eating.
(You may have already taken that into consideration, but I'm pointing it out in case you didn't.)
I know La Bresse are not known for being broody so my thought was to spread their eggs between the other breeds when they go broody and then let the broody hens hatch a couple of their own eggs as well.
That sounds sensible to me. As long as you can tell the chicks apart later, mixing breeds in one clutch should be fine.
The timing wouldn't be critical, other than my concerns about how to manage putting the roosters back together at one time.
I think putting them all together at once should be better than doing it one by one, because they are unlikely to all gang up on one. Instead, they have to settle things with everyone all at once.
If you have the breeding pens next to each other, so they still see each other, it might help. People often recommend a "look no touch" setup as the first step in introducing chickens, and adjacent pens would make that happen with no extra work on your part. So the roosters might still remember each other, and remember who is dominant over who, so they might not have to actually fight about it.
If each rooster still has his own hens, they might (mostly) hang out in their own groups instead of fighting.
Or if you put them in same-sex groups, having the pen be "new" to all of them at once will probably help, rather than having some that feel at home and see the others as intruders.