You're right; it IS a game!

Flipping a rambunctious puppy/adolescent over onto its back while it's in "fun" mode only riles them up more.

I firmly discourage alpha rolls and "dominance training". Instead, try rewarding the good behavior and discouraging bad behavior. I have a sneaking doubt that your rolling him is actually causing fear in him which would make him lash out.

There are essentially two ways to discourage bad behavior. You can use an aversive (i.e. pennies in a can) or you can ignore it. I suggest ignoring it. The problem is, you have to be consistent when you ignore it. If you ignore once and give in another time, you've just undone all your training. However, if you can effectively ignore and reward at the right moments and be consistent about it, your dog will learn much more quickly and you won't need to resort to positive punishment (which is what aversives are). Ignoring a dog when it it asking for attention is called negative punishment. I'm a solid believer in negative punishment and positive reinforcement. If you do it correctly and efficiently, you'll realize that there's absolutely no need for unnecessary aversives.