As I'm sure you've noticed, the Carter BBD has a ton of what most believe to be a bunch of extra junk attached to it. I'll spare you the technical list of parts and features and say that most are prone to failure at some point or another.
Those with the BBD also have a "computer" to help manage the function of the carb in efforts of maintaining fuel delivery. While you MAY get better mileage with the BBD, with it functioning correctly, on a stock or nearly stock CJ, it's not that much better.
There's a number of write-ups around on how to trouble shoot the BBD and even correct/repair some of the failure points. I think most that have been there, done that will also agree when I say, don't bother!
Rather than trying to fix it, long term you would be better off just ditching it for another carb. The most popular choices I've seen over the years include:
Motorcraft 2100 (Very Holley like)
Holley 2bbl (350 CFM version - looks very similar to the Motorcraft, more adjustable and user friendly)
I could go on and on about this one, but I'll stop here unless you're interested in knowing more. For the money I've spent trying out different carbs, I could have been half way to a fuel injected chevy 5.3L swap!