Do you have dual battery or intent for dual battery?
I recall there is almost no usable extra space under the hood. Is this why you asked about battery? I have a giant odessey 2150 in the lr3 2nd battery bay but can't see how 2 can be fit into the standard location - so what's the solution?
As for compressor mount, mine is an ARB single chamber, the only thing that will really fit under the hood. If one were to remove the 3rd row seats, you could do a nice mix of things under an access floor there such as compressor, large inverter and recovery gear. The seats weigh about 80 lbs and I'm guessing my stuff which replaced them is roughly equal weight: Bushranger air-bag jack, camping tables, High lift base, bag with winch chain, 2 snatch straps, pulleys, high lift (lodged between the wheel well humps just behind the 2nd row seats when folded down and ARB 40L fridge in the center seat spot. This all tucks below the floor level and also allows use of the 2nd row seats when in place.
My inverter is only 400w so it's not very big, which allows it to be mounted inside the 3rd row side-wall plastic moulding in front of the lidded storage bin. I also have 3 more 12v sockets in that 3rd row area as well as the left one on the center stack being powered by the 2nd battery.
Just to give you all some ideas
Oh, my under hood compressor fills a tank (a 2nd air suspension tank) that is mounted in the opposite location from the factory air suspension air tank seeing as there were mount points already there for it - so simple! there are outlets for it at the compressor but also inside the port side 3rd row plastic side wall 'hatch' along with the switch for the dual battery control of combined, split, etc.