Does the methane get released after the cow has finished digestion and the manure is left to rot? The carbon has to go somewhere if it's not turned into methane inside the cow.
Apparently most bovine-sourced methane comes from the burp part of the cycle rather than the fart. Manure may actually come in third when it comes to methane and the cow.
Like humans (86%!), I suspect most carbon that passes through a cow is exhaled as carbon dioxide.