Simple cheap dual battery setup

Ok everyone this is not a 4wd question but I will ask it anyway.
Firstly my 4wd has a dual battery setup with a decent size fridge. Problem is my missus does not like driving the 4wd but drives her Camry instead. My fridge is too large for her to load into the Camry plus all we need in the Camry is to be able to keep the groceries cold for an hour or two, so no replies on how to install a setup that will last across the desert.
Last night I bought a 12v cooler for $60 (new). You know the type, pieltzer (spelling) device, drops the temp by 20 deg and draws a constant 5 amps. this cooler is 24 litres, plastic and therefore suitable for the missus to carry in & out of the car. All it has to do is keep the food cool for up to 2 hours between shops.
I have a couple of 40 ah sealed lead acid batteries. My idea is to mount one in the boot & connect it as follows. + from accessories connected to one side of a relay coil, plus the make (normally open) contact, plus dioded to the aux battery +. current flow from the main to the aux battery. Aux battery connected to the break (normally open) relay contact. + of a cig lighter socket to the relay common contact.
Neg to the other relay coil contact plus cig lighter socket -. Draw this out to see what I am talking about.
The theory is when ignition is off the aux battery will power the cig socket, ie the cooler when plugged in. When ignition is on or acc the main battery will run the cig socket plus the main battery will charge the aux battery.
As the cooler draws 5 amps plus the battery is only 40 ah I am only talking about using a 30a relay & diode, 25 amp fuse in line to protect.
So shoot me down in flames & tell me what is wrong with this theory, I realise the aux battery will only float charge due to the 0.6 v drop with the diode. I don't think this will be a problem as car is driven every day & cooler will only be used once a week.
from Marty.