If the house batteries are reading 13.7 volts, they are not dead, or that is the power coming from the converter. The generator uses the house batteries for starting power, Check and clean all the connections, especially on the generator.
As for your chassis battery, it is not usually charged by the converter/shore power. With good clean connections, you should be able to use your 'boost switch' to start the MH. It would probably be wise to have the chassis battery load tested, before replacing.
Beverley and Ken