What would have happened if Maj. William Cage had obeyed General Brigham's order?

On the short run, maybe if he'd just deserted before the battle and after the briefing. On the long run not at all anyway.

So what would have happened if Cage had agreed to Brigham's orders? He would have accompanied the infantry troops as a live war correspondent with his film-team directly on the battlefield. This might have given him a little more safety than being a mere foot soldier himself, but not really much at all. Let's not forget that everyone on this battlefield, including camera teams, was effectively doomed. The whole point of the battle was for the aliens to wipe out what's left of the human resistance and the humans never stood a chance against the Mimics. Without Cage getting infected with the alien blood, humanity never had the slightest chance in that battle at all. And without Cage being exactly where he was as part of the troop he ended up in, this would likely never have happened.

The only option that Cage, who I guess we can agree was a coward at the beginning of the movie, could have made an outcome for himself that would have been better on the short run would have been to just agree to Brighams orders and keep his rank and freedom but then just pick a little less guarded moment to desert from his duties and hide from the battle. This would cost him his honours, his job, as well as his freedom, becoming a deserter on the run, but would have kept him his life. However, we still can't forget that the aliens were about to wipe out humanity and would likely have succeeded wasn't it for Cage ending up exactly where he did.

So on the bottom line it was the best for everyone, including Cage, that he acted as the coward he was at the beginning of the movie.