Maybe, but the basic problem with the engine start battery is insufficient battery charging. Forest River brand coaches do not charge the engine battery while plugged to shore power, so the engine battery can run down with just normal usage.
Your inverter may also be your battery charger, but I'm not familiar enough with your Georgetown to know. That would not be typical in a 2001 Georgetown, though.
Loss of cooling is typically not battery related. If the control panel on the fridge lights up and the switches work, it has power and that aspect is fine. I suspect that at age 17 its cooling unit is simply dying. If the temperatures inside are cool but not cold, or if the freezer chills but not the fridge portion, odds are strong that the cooling unit is simply failing and will require replacement.
