Can I just jump in here & offer some info that you may or may not be aware of ...
Been reading this long winded thread with interest, good on ya for having a go at the carb mods, albeit you've had multiple issues along the way.
Lithium batteries need a proper full charge from a Lithium spec charger prior to use, or they can never fully function as intended, also most bikes Reg/Recs especially older bikes like our Fazers were never designed to run a lithium battery, this might be where all your gremlins lie.
For this reason, I would never use a Lithium on any of my bikes, plus they have been proven to cause fires.
Might be worth swapping out the Lithium for a fully charged lead acid battery & see if there's an improvement.
You've obviously got other gremlins to diagnose, but Mike as always has given you the heads up on what to check already.