I'm thinking that if there's enough juice to get her started once but, the alternator is not supplying enough to keep it running. The battery is flat, having used everything to get it running so, there's nothing left for spark... It sounds like an alternator (remember one battery is new).
I'd still put a battery switch in while you're there.
right on, think I took it as his alternator is fine, because:
"You can turn it off and on while on the lake with no trouble at all"
if thats the case, its battery or connection.
if it was mine, Id stick the multimeter on it, make sure the voltage is high enough while running.
then check it again sitting.
I just had a head scratcher on my wife's pathfinder...Its had a couple hiccups lately, like the key gets left on ever so slightly... then its dead 30 minutes later... that happened a few weeks ago twice, stranded us once. On the way to town, the ENTIRE electrical system dropped out for about 3 seconds. it was BIZARRE. also, I had an issue with the batter connection being loose. SO I thought, I'd fix it that evening. didn't make it.
the battery just ate it HARD. and fast... like it would show voltage, and charge fine, but as soon as you flip the key on, it drained immediately. It was weird.
it had no amperage left. PLUS, the positive terminal, was worn down, and wouldn't clamp tight, I had assumed it was the clamp.
Batteries are weird, and the more electronics you throw at them, the stranger it becomes.
another time, on a yamaha raptor 700 (EFI) a friend had a very heavily modified engine...we went for a night ride, and it just died. Battery so weak, not even the charging system, which tested fine, could keep it alive enough to run the electronics.
that seriously sucked. lol