Is the coolant reservoir near that area could it be leaking? The way an automotive shop would locate the leak is by removing the radiator cap and attaching an adapter that screws on in place of the radiator cap. Something that looks like a bicycle pump with a pressure gauge attached to it is used to "pump up" pressure to simulate pressure generated when the cooling system is at operating temperature, the leak is then found sometimes it is obviously leaking from a coolant hose or cracked housing, failed gasket or water pump, sometimes the engines cooling system needs to be inspected from below or inspected using a mirror and flash light. If the coolant loss is still evasive dye can be added to the cooling system that may indicate the source of the leak. A mysterious coolant loss may even be a sign of a head gasket that is not sealing well.
have you cooling system pressure tested. 1 gallon in 10 miles is quitea significant leak and should be easily traced. It is possibly the radiator. Coolant leaking from the radiator will collect on the crossmember and drip on either side depending on how the vehicle is sitting. There are no coolant hoses under or near the battery.