Is anyone familiar with the Carter BBS one barrel carb used on a 1963 Dart?

Not completely familiar with your carburetor, but I had to recently rebuild a Motorcraft 2150 carb circa early 70's, the accelerator pump on a lot of the old carbs are an endless source of frustration, namely it's the gasket for those, and good luck finding a rebuild kit for them, they never have the parts you really need. While I can't diagnose your problem, I can suggest it is extremely well worth it to have the original carb professionally rebuilt, make sure the timing is good, and of course all springs are hooked up correctly. Wouldn't hurt to also confirm good fuel pump pressure.

these old carburetors have small ports and metering valves to mix gas and air under different conditions , the float level is one small detail ,,a hung up accelerator pump will cause problems , gaskets in a kit that look the same may have one small difference , .Or you can buy an after market one or a rebuild that you can "bolt on ". .

Sure the float is "floating?" Also- sounds like accelerator pump to me... If it's flooding, you'll see that on the spark plugs easily.