Stop pulling numbers out of your arses, people.
You can approximately figure the peak power consumption from the recycle time. Specs say it's 4.4s and the flash energy is 4800Ws (4.8kJ), that means you need 1100W... plus inefficiencies so say 1500W on average while it's recharging. It might be uneven, e.g. a big initial peak of 2kW or 3kW that falls off to lower numbers near the end of the recharge - it depends how smart the charging circuitry in the flash is. You must also add on the total power consumption of your modeling lights (less so if you set the pack to turn the modeling lights off while recycling); the pack supports up to 2kW.
A small 5kVA generator will not struggle with this at all, even with 4 of 500W modeling lights attached and running continuously. A 2kW inverter will probably be fine with it if you turn the modeling lights off.
A small car battery (e.g. for a 2L car) can supply about 300A (3600W) instantaneously, but only for a minute or so. Keep in mind that the capacity of a battery goes down rapidly as the load current goes up, which means that if you can halve the load current, you can probably get double or more total capacity from the battery. This is why there are power-packs designed for use with batteries; they charge more-slowly so as to maximise battery capacity, and they do direct flyback switching DC-DC conversion from 12V (or 48V or whatever) to 300V for the flash, no need for sinewave inverters, PFC and all the inefficiencies involved therein.
If you want to use several car batteries, be aware that connecting them in parallel will cause significant current to flow between them if their charge is not perfectly equalised. Keep your batteries as a set and only disconnect them from each other for transport; leave them connected to each other at all times during use and charging.
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