My last mentor had a 2006 or 2007 KW T2000. He had a mini fridge in the truck, and not sure it was something he put in himself, or if it was standard equipment. Either way, he told me on the first day i was in his truck, "...this switch here, powers the fridge...it never gets turned off..." And for the 4-5 weeks i was on his truck, it never got turned off. and we never had a problem with the truck not starting.
As far as your inverter goes, one thing i remember from when i worked in a truck stop, was the shop guy there telling me that just because you had a 1000 watt inverter, dont mean you can run several items that would total 1000w. You will greatly reduce the life of your inverter, and might pose a fire risk. If you have 1000w...i'd run it up to about half capacity. You will get the most use out of it, and you wont have to worry about fire..................unless the guys who install it are complete idiots.