[Cthulhu_]

I googled quickly and the summary says "A single cow on average produces between 70 and 120 kg of methane per year", so that would actually probably be feasible. Getting it would the challenge though.

That said, if I were to over-engineer this, they could make barns with a closed air system that filters out any methane produced.

[NotSwift]

This might not be as far fetched as you assume. Large farms in the Netherlands are already required to have some filtering systems for reducing the output of Nitrogen-based compounds.