On the generator sizing issue, some very basic battery chargers draw large peak currents at the peaks of the AC waveform. Some generators cannot handle this well even if they are capable of delivering the required average power.
PS: Another factor is that if the battery charger contains a step-down transformer (most will) and it is relatively lightly loaded compared to its full rated output the transformer primary winding will look like a large inductive load in parallel with the resistive load associated with the output to the batteries.
Highly inductive loads can also cause problems for some transformers, and will also require the generator to be able to deliver more current than you would expect just from the power going to the batteries.
PS: Another factor is that if the battery charger contains a step-down transformer (most will) and it is relatively lightly loaded compared to its full rated output the transformer primary winding will look like a large inductive load in parallel with the resistive load associated with the output to the batteries.
Highly inductive loads can also cause problems for some transformers, and will also require the generator to be able to deliver more current than you would expect just from the power going to the batteries.
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