Having larger paver type blocks under the piers will provide better floatation and slow settling some, I did this with my tool shed but I used 16" square patio pavers with standard foundation block on top of them.
A 2x4 screwed on the bottom would help but wouldn't be overly strong, with a smallish coop like you are doing I would put 2 or 3 2x4s nailed together standing up on edge to make a laminated beam, then support this on each end with your piers, this would give you more than enough support in the center and really strengthen you center joining point of all the pallets as they would then be setting on the beam rather than just floating in mid air with a 2x4 nailed on the bottom. Your idea of a 2x4 would probably work as it is just a coop and chickens and coop contents are light enough to not stress the floor too badly but it may be springy with people walking around in there.
A 2x4 screwed on the bottom would help but wouldn't be overly strong, with a smallish coop like you are doing I would put 2 or 3 2x4s nailed together standing up on edge to make a laminated beam, then support this on each end with your piers, this would give you more than enough support in the center and really strengthen you center joining point of all the pallets as they would then be setting on the beam rather than just floating in mid air with a 2x4 nailed on the bottom. Your idea of a 2x4 would probably work as it is just a coop and chickens and coop contents are light enough to not stress the floor too badly but it may be springy with people walking around in there.