Been there, done that, with the Honda Eu1000I generator, which in your case, is the limiting power factor here, the weakest link.
You really need 14.8 volts, period. The Progressive Dynamics 9245 at 45 amps, runs fine on the Honda EU1000i, but it runs to 14.4V with the pendulum added. It's amps drop off quickly, but it will make the little Honda work pretty hard up front, so warm the generator up first, before plugging it in and running it in regular mode for a bit before plugging in the charge controller.
A better option would be the Iota DLS30 with the IQ4 smart controller, which will give you 14.8 Volts in bulk mode.
No matter how you slice it, when you are running on a generator, to a limit, about 14.8 to 15V.... Voltage with a modicum of amps pushed behind it, is what it takes to save generator run time, without overloading the generator.