Hi @philipp
hope I didn't misunderstand your questions
by default weight must be like this https://forums.flightsimulator.com/t/black-square-steam-gauge-overhaul-king-air-350i/548884/789
I suspect something wrong in your install, maybe try a uninstall via my PC -> application control panel -> uninstalling application then reinstall about the battery load, I m not really aware of it s capacity and load, just know it s fast drained
you can search if it always has been answered then ask on the official msfs forum since the thread is pretty active
regarding you pushback issue what are you using ?
try using shift + P shortcut with parking brake released shouldn't be an issue no matter the battery estate
hope I have helped