I use the same type of feederhi
The chicken part of the coop is 8’ x 8’ and the covered run is 10’ x 12’. We have an auto door between the two that opens 1/2 hour after sunrise and closes 1/2 hour before sunset. I was worried that they wouldn’t have food or water while they are in the coop if I put it in the run. Attached is the feeder that I am using.

Birds at roost dont eat/drink (really they dont move....which is why so much poop accumulates under roost spots). Before going to rh oost, they fill their crops... overnight they digest that store of feed.
There is no need to have feed or water in your coop given the details you provided.
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