Looking at the extremes doesn't tell you that chicken production is an increasing-cost industry at the margin. Sure input costs are important (the OP agrees - see last section), but there are also economies of scale, R&D investment, and so on pushing the other way, so it's ultimately an empirical matter whether chicken production is increasing- or decreasing-cost at current levels of production (again I'm just repeating what the OP says).

IMO this issue is actually less relevant than the OP seems to think, because we're only talking about very small marginal changes to chicken demand, and there's no way the long-run supply curve is steep enough for that to matter. But one could try to estimate the long-run supply curve at least "locally", which might settle this issue.