You might want to investigate what your local rules and laws are about keeping chickens. Each locality has their own laws and restrictions, whether that is a state, county, town, or HOA. That way you are prepared if someone does complain. There may be something you can do that would make you legal that you don’t know about. I just think knowing what you are dealing with is a good thing. Not knowing leaves you vulnerable.
As to what kind of flock you’ll have with BA, BR and Productions Reds with a PR rooster, a flock that lays pretty well. The genetics for laying are passed down by both the mother and father. Production Red is a generic name, more of a marketing name, certainly not a breed. I wound up with some and they were a little smaller than normal dual-purpose cockerels but different hatcheries make Production Reds different ways. Yours may be bigger. I put mine in the freezer a couple of days ago, they’ll still make a nice meal even if they were a tad smaller than some others. Production Reds are generally really good at laying so they will have those genetics.
With that rooster your first generation of chicks will be either black or red. The sons of your BR will be barred, a black sex link. But if you don’t save a barred rooster at some point barring will disappear from your flock. Barring is a sex linked gene, mothers only give it to their sons but a rooster gives it to both sons and daughters.
If you save a rooster from a red/black cross (BR is basically black with barring added) and mate him with pullets from a red/black cross it starts to get interesting. You’ll still get a lot of black chicks but your red chicks will be various shades and patterns. You may get golden reds, you may get dark reds. You might even wind up with mostly white chickens. If you save a barred rooster you might get red barred chickens. It all depends on what genetics your original parents had and which chickens you save for mating. I think it would make a pretty flock. And with those parents it should lay well. If you save larger roosters you can even get the cockerel butcher size up.
Instead of keeping a rooster you can get fertile eggs if a hen goes broody. If you look on the Indiana thread in the “Where am I? Where are you!” section of this forum you can probably find someone local with fertile eggs. They might even know what your local rules are. You’ll still hatch out cockerels so you have to have a plan for those. There are just so many different was you could go.
Good luck!