On a frictionless road and tires, I know the rider wouldn't be able to change the combined center of gravity of the system by moving his weight around. If he pushes the bike to lean to the right, he would go to the left by the amount that would keep the combined cog the same. But a bike with real tires and a real road under it is not a closed system, the tires wouldn't let the lower part of the bike left or right when trying to initiate lean.

Does this mean it is possible to push with your bodies inertia to the bike, and the combined cog will be changed?

I think if you do so below the center of mass of the bike, you could, but I am not sure, since the tires will want to move more, but won't be able to, making the combined center of gravity shift more than if you applied the force higher up. And I think you should use horizontal forces on the pegs not up/down forces on the pegs

I know counter steering is the best way, but I would like to think it is possible to change direction without it, without the influence of handlebars

Please ignore gyroscopical forces and assume the bike doesn't have any trail geometry (so you would have to "catch" the bike manually through the handlebars)

Thank you very much