We all know Grant is cheap in either version (Japanese or overseas). In the overseas version he has his crazy attack speed. In the Japanese version, he has his crazy range AND crazy attack speed (he can only throw one dagger at a time, so the closer an enemy is, the faster he attacks). Coupled with the fact his Doppelganger throws an axe over his head, Grant's a cheapskate in the Doppelganger fight.
But he's not the only one!
I got this glitch to happen on Trevor. Not sure if it would work for Sypha, but it might. I know it won't work for Alucard. When you see the Doppelganger, run toward one of the ledges on the left or right. You have to time this just right or it won't work, since his AI is scripted to jump straight up if you're too close to him, and if you time it wrong even if he jumps toward you he might land short. Basically you lure him into jumping onto one of those ledges and then hide under the ledge. He can't hit you! As long as you have the Axe or Holy Water on Trevor, or the water or ice on Sypha, you can attack him without any risk!