What is the best way to level Co-Op Score in Battlefield 3?

As for me - easier level is Fire from the sky - safe and could be very fast with good flight skills. With known friend (not a random pilot\gunner) and voice chat you can split attack areas and eliminate npcs much faster. Average I got 12k per flight, dont remeber how long it could take, but leveling for unlocks was pretty fast. projecting on the many repeats - 20% more exp and guaranted safe completion worth this.

I don't think that there is an easy way to work your way up to the dog tags.

All of the ways require work (unless you want to cheat which may cause you to forfeit your progress entirely). Your best bet is to find a level you are comfortable with playing and find a capable team mate to help you do so. If you become an expert at a level then you may be able to find ways to speed it up by coordinating with your team mate or skipping unessential combat.

What is unessential combat? If you see your objective to the west but enemies to the north, you can avoid them if you have good cover leading you closer to the objective.

You could also speed up the time by becoming efficient with your guns (obviously). If you can aim down sight and engage an enemy quickly and accurately, you will split time off of the mission. Conserving ammo also helps as you wont be screwed over when you have to reload in a hectic combat situation.

Getting to your goal isn't going to be easy. If it was, it wouldn't be special or worth a celebrating.