I'm afraid it's not a dietary thing, but a behavioral one. And there's no way to keep him from doing it other than limiting his outside time to supervised, on-leash outside time, as he's probably scarring his poor inestinal tract with the sticks!
See if you can get him some very crunchy - but Cat-Digestable - treats, to give him instead of the sticks, which he probably likes the texture of. Did the vet check his teeth?
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