Chuckie chicken
Chicken Chucker
I usually get my inspiration from endless staring at the materials I have on hand. 
Could you mount a guillotine style door on the inside of the 2x2's so it doesn't have to contact the hardware cloth? Your metal skirting would be a good shield against coons lifting it from the bottom. Plywood thickness wouldn't be a concern. To keep weight down, you may be able to frame a pice of metal siding to use as a door, although that would obviously be more work and more fiddly.
For an exterior mount, it could rotate open and closed from a single bolt in the upper right corner and slide into a channel on the left side and bottom to secure it. You may still get that nasty chafing from the hardware cloth.
Just spit-balling
The little coop looks good!

Could you mount a guillotine style door on the inside of the 2x2's so it doesn't have to contact the hardware cloth? Your metal skirting would be a good shield against coons lifting it from the bottom. Plywood thickness wouldn't be a concern. To keep weight down, you may be able to frame a pice of metal siding to use as a door, although that would obviously be more work and more fiddly.
For an exterior mount, it could rotate open and closed from a single bolt in the upper right corner and slide into a channel on the left side and bottom to secure it. You may still get that nasty chafing from the hardware cloth.

Just spit-balling

The little coop looks good!