If they wanted to attack GMG, invading Alpha instead of capturing the Taus and attacking Bretonia is way more feasible and easier. Hell, even IMG would actively support the attack against the sniffers. Then the Sigmas would fall to Charles' hands if he was able to overcome the guerrilla tactics and the stretching of his supply lines (Gallia -> Sigmas) without the need to attack any House but instead letting them fight their own wars.
