I have seen pics of the stage coach & I understand what it is you speak of...
I think you may have 2 other options.
#1) there is a wall where the chickens roost at night that faces the end of the coach where the feeder is....
now I'm not a woodworker by any stretch of the imagination but can you build some kind of a pan to go under the birds into that wall??
putting it here, you would not have to deal with the wire or the cover.
#2) vinyl flooring. where the chickens roost at night, why not consider vinyl flooring? it does not have to be glued down(lest you prefer it that way), and it's easy to clean because as long as you keep up with the birds, nothing will stick to it like it will stick to wood.
The vinyl can be removed every time you clean, hosed down, and then after it dries, put back in.
Again, these are just some ideas I was thinking about. I have a full size permanent chicken coop that does not move & I have 2 layers of vinyl in there with the birds. Clean up for me is a piece of cake(so to speak). I have 75 birds that I take care of & spoil rotten.