Edit: also they take a lot of land to feed. Soy grown in where Amazon rainforest used to be and so on.
And on the methane front, they eat the carbon in the grass which has literally just captured a bit of carbon from the atmosphere. They fart it up and then it comes back down in a reasonable period of time and goes back into the grass they are grazing upon.
The cycles and definitely complex and opaque but we need to get better at working with nature rather than engineering new ways to ignore the real issues.
It actually doesn't matter if you grow the feed nationally or import from Amazon, the fact remains that those fields replace a natural ecosystem that could be contributing to biodiversity and much more efficient carbon capture.
Grass captures CO2 from the air, and the cows fart (actually mostly burp) out methane that is about 25 times worse GHG over a 100 year period.
You want to get better at working with nature? Hunt your own meat from the wilderness.
Look, the problem is real simple. Simple math says that we can do this:
Carbon(in atomosphere) + Carbon(biomass) + Carbon(underground) = Carbon(total on earth)
Biomass gets it's carbon from atomosphere, and releases (most) of it back to atmosphere, and hence not a real problem as it undergoes a stable cycle and is limited by the total carbon between the two. Even if it's form changes, it's not a real problem as the math shows it's inherently limited. Additionally, some carbon does escape this cycle and ends back in underground stores - but it's an extremely slow process.
The real problem is taking carbon from underground, and putting it into the atmosphere - via an unnaturally rapid process. Anything else, is comically trivial to the problem that is the fossil fuel industry.