If only the volume of air could be measured like water we could place a baloon type device in the empty compartment and inflate it to fill all the voids. Any thought along these lines?
Get a 'Balloon', it will need to be a fairly tough one which when fully inflated is quite a bit bigger than the space you are measuring.
Place in glove box (or whatever).
Almost close glove box, leave open just enough to either pull the 'valve' of the balloon out or maybe put some small dia tubing to the balloon.
Inflate the balloon to just over atmosphere. You want it to take up all the room but not build up much excess pressure. It may be better to evacuate the air from between the outside of the balloon and the walls of the glovebox but making a seal like this may be difficult.
Seal balloon.
Pull out the 'inflated balloon'.
Imerse in water, the water needs to be up to the top of the pan or spigot. You need to make sure and not immerse it too far or the water pressure will compress the air. I'm thinking a large but shallow pan. Maybe push the balloon down with a large board or something so it can spread out but not be submerged too far.
Measure the displaced water.
I'm pretty sure I missed something.
If the water insists on compressing the ballon you could put a pressure gauge on the balloon to see how much and compensate to adjust the volume.
The balloon wont actually be a ballon, I'm thinking something like the heavy duty bags they use in shipping or something.
Sounds mighty complex, someone must have a better idea.
KENAT, probably the least qualified checker you'll ever meet...