A hundred watt load at 12.2v DC is 8.9 amps. With 80% inverter efficiency this load then becomes 9.82 amps, on the battery.
A group 27 battery, which is likely larger than that which came in a Minivan, is usually around 100Ah.
Somple math would have the healthy new fully charged group27 battery, under a 100 watt load powered by an 80% efficient inverter, depleted to 50% in ~ 5 hours.
But simple math does not account for the peukert effect, which basically says the bigger the load the lass capacity the battery has.
The 100Ah battery earned this rating by supplying a 5 amp load for 20 hours until voltage fell to 10.5v, which is considered 100% discharged. The 100 watt load on an inverter almost doubles this load on the battery , and according to this Peukert calculator, the 100 Ah battery under a 5 amp load becomes a 70Ah battery under a 9.82 amp load. That peukert number used is that of a high quality AGM battery like odysey or Northstar, so on a OEm starting battery that capacity is likely around 63AH.
http://www.csgnetwork.com/batterylifecalc.html
So that 100 watt load powered by the inverter, can deplete the healthy fully charged starting battery( which do not like to be deep cycled) to 50%, in about 3.5 hours.
And this is likely also generous, as as the battery depletes, its voltage drops, but the inverter is demanding the same wattage, so it draws more amperage, as volts times amps equals watts.
The point of this?
Inverters are excellent battery depleters.
Secondary point, is that vehicle charging systems are not magic. While an alternator can indeed provide large amperage when it is asked to by its voltage regulation and there are suffieient loads to consume those ampheres, it cannot force a battery to recharge to full quickly.
One can easily drain a 100% battery to 80% charged in 10 minutes with enough load. One Cannot recharge that 80% battery to full in less than 3.5 hours, and that assumes a fairly high amperage charging source that is seeking then holding ~ 14.5 for those 3.5 hours.
Which NO Vehicle will allow.
Except mine, as I am the voltage Nazi, and 13.7v when the battery is still less than 100% charged, was not tolerable.
But the rest of you are up shit creek in a barbed wire canoe.
Consider the parade, rained upon.
Just kidding, congrats on the new minivan