As a right handed shooter, I think it would feel more natural to move my hand from the grip to operate the sidelever on the right side, while controlling the opening of the barrels with my left hand, which remains on the forend.
This would also prevent wear and damage to vintage doubles that do not have adequate joint checks. We have all seen that picture of the Fox shotgun with the cracked frame that the Preacher has posted here about a thousand times to demonstrate that they are prone to breaking. In reality, this isn't happening due to firing the gun. Dewey Vicknair has explained that this frame cracking happens when the barrels are permitted to drop fully open, and the hammers stop abruptly when they strike the inside of the action flat. In short, it's a matter of abuse by a careless shooter that causes Fox frames to crack.
