Why would you need a new carb? It's such a simple device. I had carb trouble with my BF8A and I took it completely apart and soaked it in a quality carb cleaner (from a Ford dealer) overnight and every part, including the jets were completely sparkling clean. The motor ran like a top after that.
I cleaned all four carbs on my BF90. It was pretty easy. Removed jets, cleaned with Seafoam and Mass Airflow Sensor cleaner, then used compressed air liberally.
The air screw is broken off flush with the recessed brass insert. That is the reason . Or maybe someone can tell me how to extract the broken part, its maybe 1/8 inch in diameter. Don't think its possible to drill in the center of this and use an easy out if you could find one that small.
Yes, they make easy outs that small and that is what you're going to need. I'm also a gunsmith, so I have to use small easy outs from time to time. You should be able to get a set through Brownell's.