It's your carb. I think it might be the float sticking..? Mine did this several years ago and putting a new weber on it fixed the problem. I rebuilt the bbd 3 or 4 times and was just sick of it. I think my mechanic said the float was hung up and keeping gas from getting into the jets, but it's been a long time ago so I don't know if that makes sense or not (I don't know much about the bbd except that it's a piece of crap) Might be the fuel pump too. If the gas is running back down in the lnes all the way to tank, it would cause the jeep to be starved for gas until you turn it over enough to get the pump to pull it back to the carb, right? Just something else to look at.
85 CJ7 258, borla header, Team Rush, MC2100 ,optima batt, Warn 8274, agr super box 2 & pump, T5 w/ B&M shifter, centerforce 2, superior 1 piece axles, warn hubs 2.5 in lift, 31" BFG MT
85 CJ7 258, A/C, T176, MC2100, Team Rush, Warn XD8000i, superwinch hubs, 3in lift 31" BFG AT/KO