The fridge is powered by the house battery. The engine is started from the chassis battery. The fridge should not be causing the chassis battery to run down.
If the motorhome had not been run for awhile last time, the chassis battery probably was weak just from sitting and from the usual slight constant loads on it. Typically the chassis battery is not charged by the converter or anything else when plugged into shore power. For long-term storage, it's best to disconnect the chassis battery or provide some means of keeping it charged (a small solar charger, a float/maintenance charger, or a gizmo such as the Trik-L-Start that steals a power from whatever is keeping the house battery charged to keep the chassis battery up).
