I maybe mistaken but I am pretty sure your house batteries do in fact charge off your alternator while the engine is running. I think you will find your chassis batteries charge first then after your chassis batteries are up to around 13.6 volts the Trombetta relay energizes and now the house batteries charge also. Look at your monitor panel to see what the house battery voltage is before starting your engine, after driving a while check it again and I think you will see a higher voltage.
Back to your post I think you will see lower voltages in warmer outside temps.
Blessings and Merry Christmas
