Hes here ...
you could ... its seems like a lot of trouble to me ... the reason you want to run the pump from the van battery is because you dont want to run two cords to the house ? its not much more hassle to run two as to run one , or tie two extensions cords together ?
also you'd have to look into the running time youd get on the pump from the battery , youd need a second battery on a split charge system ... than a 50AH lasts 6hours using a 100w device ... so if a Storm 800psi half HP pump is about 350w ... would last a couple of hours on paper , perhaps not last a full days cleaning without charging again .
Was never fond of the vanmounted portable setup myself , most of the time its just easier to pull it out and get close to the job . But it suits some i suppose .
What machine are you considering .