Actually same. The one cart repulses form another through spring. It does not what spring shape is. The spring must have same forces on both sides on ideal model.
I see. Yea I realized afterwards that it doesn't matter about the details of the spring.
Still, you need to determine errors and control for factors that might make it unfair. For example friction. I know the friction seems low, but the friction of a swivelling office chair also seems low, yet you can propel or rotate yourself on it without pushing against the desk.
You seem to be confusing force with energy. The cart with the wheels receives more energy. It ends up with translational as well as rotational energy. But it doesn't receive more force or more implulse - those must be the same, as you said.
How can you have more energy with the same force and impulse? W=Fd. The cart with wheels has the force applied over a greater distance. That give it more energy with the same force.
I wish you could be clearer on what exactly you think is wrong. Do you mean the law of translational momentum conservation is wrong? Is it an internal contradiction, or disagreement with experiment? Don't say both. We only need to focus on one.
