I have not come up with any such information in the various on line calculators. Your experience will be greatly appreciated.
That's exactly what information they provide, that and figuring out how long a wire is after you wrap it around something.
I don't see that you're saying you tried 2 of the exact same coils and that worked fine, but 4 doesn't. THAT would be a battery problem.
According to Steam Engine, it would take about 20 watts to get one of those coils up to a minimal heat flux. That would take about 6.3 volts. 8 volts from series batteries is too much, that's actually 61% more power because power goes with the square of voltage. Series batteries want to put out 4 times as much power into a particular resistance, due to this square law. You would need coils twice as long as for a single battery, which would not be higher heat density but twice as much heat total, as they're twice as big.
I see some talk of bigger wires taking longer to heat. While this is slightly true, it often ignores the question of whether it will
ever heat up -- if the heat flux isn't right, subtleties are moot.