it may be your adapters. put your 510 or 801 adapter on and put the tip of your tongue on it when pressing the button. this will let you know that you are getting current and that there's no problems with your center post, batteries, or anything below the atty. don't worry about putting your tongue on the adapter, it's nothing harsh at all, just a slight tingle. if you get that tingle, then connect your atty but don't over-tighten it. particularly with the 510 attys, you only want to feel the resistance when tightening it, and then turn it just a bit more to line up the air holes. over-tightening them can cause various problems, like a short circuit, particularly with aftermarket adapters too. if these attys are working on other setups, this should remedy the issue with your GGTB. if you're not getting the tingle, then the issue is below the adapter. take it off and try the same tongue test with the native 901 connection. if that works, the problem is the adapter itself. if that doesn't work then the center post of the native connection cold be the problem, or the batteries, but this is probably not the case. failed adapters or over-tightened attys, particularly 510s, are common problems...hopefully that's the case here.
as far as the attys at 6V, well the GH1 510 (Boge I think) is 2.8Ω, which is doable. even the standard Joye 510s at 2.3Ω are doable at 6V, but that's a lot of watts, a lot of heat, and their life and performance is going to diminish a lot faster than a higher resistance atty would in the same setup. i don't know what the TW 510s are, if not Joye's, but at 6V use at your own risk. same with the 2Ω 810 atty. you'll want to pulse the button, rather than hold it in, keep the atty wet, and take shorter hits as to not overheat it and pop it prematurely. i use that 2Ω 801 mini at 3.7V in the GGTB with the AW17670 and it's a nice vape for me, but i can appreciate going even lower and generating more heat and a harsher throat kick with a LR. however, 2Ω atty at 6V may be a bit much, and even with the standard Joye's i'll only use them at 6V in a tank system like the AVS/AFS.