You just need to do the maths P=VxI for the output therefore
240volt @2 amp = 480watts ok
@2.25amp = 540 ok
@2.5 = 600 inverter on MAX
@2.6amp = 624 inverter overloaded (alarm)
@3 amp = 720 inverter overloaded (Alarm and shut down)
I have seen 2, 3 and 5amp rated battery chargers
The input is harder to work out due to the inverter type and a few variables come into the mix like 85-90% efficiency cooling fans etc. Running a 600watt unit, just check the input fuse fitted/or recommended and that should give you an idea of the Amps at maximum load so if the unit is fused at 35amps and used on a (NEW)110aH battery it could last 3.1328 hours (half the time required to charge a flat Ebike battery) unless you run the engine, though the inverter might auto shutdown before this on its low volts threshold (usually 10.5volts). But normal car batteries are not meant for prolonged high current drain, they are usually 40 to 50Ah at the 20 hour rate. And the 16mm cable you are using is rated @ 61 amps though a long run will reduce that with a volt drop @2.8mV a meter.
NOT ALL CHARGERS ARE THE SAME – check the specification on the back and the size of fuse fitted (especially the MADE IN CHINA ones!) or check with your supplier and see if they can confirm the charger details and if the charger is ok to use on an inverted supply to charge your battery, some are not! as they use sensitive electronics to detect inductive resistance to gauge the battery charge condition. This is possibly more relevant with the newer battery technologies with BMI circuits built in. Just because one works does not mean yours will and I would not like to see you stranded in the wilds of Italy with a bunch of flat batteries as I know how hard those Ebikes are to pedal up hill
