Still doesn't answer why you don't get a big (car or boat type) battery instead of these convoluted power station thingies...?
Internally these also just have a battery, plus some electronics that convert the battery's DC to AC, only for you to plug in a PSU that will convert this AC straight back to DC again with which you then power your DC charger. You pay extra for converting power twice (and inevitably losing some in the process) while ending up with exactly the same type as you started with (=DC). So unless you need that AC power for anything else, it's indeed kinda crazy to do it that way.
So why indeed not just get a battery instead and hook the charger to that directly? You don't have to use the one in the car you drive to the track/field with (but it
is a valid option, unless you plan to drain it completely, of course). You can get a simple battery cheaper than these power stations, you don't waste extra money and power by uselessly double-converting electricity. Total power output should not be as much of a limiting factor as well and larger capacities should be available, too. Although capacity will of course drive costs up.
Not even the Amp input limit of the 4010 of 65A would matter on a battery, as even with only a 12V input, that still ends up with max. 780W - comfortably enough for your intended 15A charges (~630W, remember?). Only if you wanted to use even more Amps, 12V would no longer be enough anymore and a source with higher voltage would be required (e.g. two, three or even four 12V batteries in series would work, as the limit for input voltage is 9-50V on the 4010.)
But maybe you should start looking into some kind of generator/alternator instead, if capacity is an issue.