Bart,

I preach the KISS principle: Keep It Simple Stupid.

Your inverter should be OK without the second battery but grounding it at the back of the vehicle has the POTENTIAL of minor electrical gremlins. Try it that way first but if there are problems ... then run a ground to the battery.

A second battery will give you a lot of peace of mind. The set up erland mentioned is the most user friendly but more complicated for install. As an alternative, you can install an inline switch & a 5 ohm resister in line between the batteries. The switch will allow you to isolate the rear battery while camping and the resistor will allow the rear battery to charge without drawing down the altenator.

I don't think there is any need to upgrade your altenator unless you are installing some workhorse electrics like a winch or enough lights to outshine Vegas.