Is it possible to destroy Dragon Mantle without the owner successfully activating it?
He cannot prevent you from pumping it. As you say, you can either pump on your own or pump in response to the destruction, either way the pumps will resolve and his creature will die. The only way to prevent this is to kill the creature it's attached to.

If the enchantment was to read "R: Enchanted creature gets +1/+0 until end of turn as long as ~ is attached to it" then things would work as your opponent would want. Obviously this isn't the case, and I am not aware of a card that functions in this way.

Similarly, if he has priority and you have not pumped yet, he can prevent you from activating it with a card like  Krosan Grip.