sounds like water in the fuel, as you slosh it around under way, the water on the bottom mixes with the fuel on top, and shows when the engine is under load and need the fuel for power. Idle is just the engine free wheeling, and can handle water and not effect it.
Rig a temporary tank up and try it.
As far as your electrical issues, I suspect you blew the fuses when you disconnect the batter while running, as alternator tend to spike when that happens.
Electrical engine problems are rarely intermittent.
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