The new ones are 400 watts, so U not gonna run much, dang sure not yer toaster, LOL.
I'm a retired Ram/Jeep engineer that was in charge of the battery/starter/alternator system. If you need more than 400W you go directly to the battery terminals, or on later models the "HAPP" (Hi Amperage Power Point) if so equipped - and then use an aftermarket inverter.
Alternator size is not correlated to the 400W outlet.
The Ram Upfitter web site has alternator output curves that will show you how much alternator capacity is available, minus whatever accessories are turned on and what's necessary to power the engine electronics (12-20 amps). Alternator output capability is totally dependent on engine RPM and underhood temperature.
If powering off of the battery(s) alone the available energy is totally dependent on the battery temperature, battery age, battery state of charge, and the rate of amperage draw.
FYI - recharging the vehicle battery by simply driving or running the engine will take longer to recharge than using an auxiliary charger since a good sized charger will provide a higher recharge voltage that the vehicles alternator which has a regulated output voltage curve. If the vehicle has a start/stop system that uses a second small AGM battery then higher charge voltage becomes more problematic if too high.