There’s a limitation due to a complication that people are avoiding noticing. Kinetic energy and momentum are two different things. The OP talks about vectors and setting the length to zero, but the “energy” (kinetic energy) is a scalar value, not a vector value at all. The linear momentum, on the other hand, is a vector quantity.
Consider the example of two billard balls colliding. As a model of perfect collisions there is a unique solution to how the balls roll, because only that maintain (the total of) both quantities. It is limited in the sense that one ball can’t just convey an arbitrary amount of kinetic energy to the other.
Second, momentum vectors can combine from different directions to cancel out. The idea of storing and then calling up does not work as you suppose. Let’s say he pulls out 50 units of momentum pointing North. If he had near 0 to start with, then now he has 50 units pointing South left over. So really he can conjur up any momentum at any time and what’s left in his internal account is just for satisfying the physicists that momentum is globally conserved, and has no observational purpose otherwise.
If you want otherwise, then you need to impose a limit on what he can store. But note again how vector subtraction works: It is a limit only if he wants to produce momentum in the direction opposite his current store, and he can reset that by producing momentum in another direction.
In other words, his ability to not show any recoil will be limited. If he wants to keep throwing stuff “that way” then he needs to balance it out with throwing stuff in the opposite direction too. Or just wait 12 hours in the case of due east or west!
Now his internal kinetic energy store can behave in a more straigtforward manner, since it’s a scalar and not a vector. He needs a positive stored value to draw down, to make his power work, and it’s depleted when it dropps to zero. In this account, direction does not matter.
So how does he recharge that? Shouldn’t stopping an object consume kinetic energy from his store too, rather than replenishing it (like regenerative breaking)?.
Now that you understand the need for two separate accounts of stored physical properties (one vector, one scalar), we come to the real, interesting, limitation.
To manipulate objects of various masses, he needs to use both values in a controlled manner. First, you might consider limitations of his control or of the underlying ability to transfer the values. This happens in real life when a hammer bounces back rather than delivering all its momentum and energy with the blow. If his power works “like a hammer” at the point on the object he decides, it may bounce back some of the momentum. This limits how fast he can affect the object, based on its composition and physical properties. It might also, like with using a hammer, hurt and tire him.
So, a large mass covered with padding may be impossible to push rapidly enough to change the situation in his favor.
Using the hammer analogy and confining application to the surface, it is easy to explain difficulties and limitations in manipulating objects.
Second, the momentum and kenetic energy accounts are separately limited in what he can store. This means that objects that exceed one or the other would be a problem. A small, fast object will exceed his kinetic energy store, and he’ll have to shed it by throwing other objects away, until he runs out of other close-at-hand objects or runs out of time.
A huge slow object will overwhelm his momentum store, which he’ll have to shed by throwing something else in the same direction. But it’s providing far less kinetic energy, in proportion, so he’ll not have energy enough to shed the momentum by throwing small items. He will be crushed.
Finally, back to the hammer analogy. If he delivers the dose of energy and momentum off center, it will impart a rotation to the object being controlled. And angular momentum is not among his powers. So he will feel a torque recoil from this, and set himself spinning.
What happens if you bounce a spinning ball? If the transfer is not instantaneous and behaves in a similar way, the adversary could throw spinning objects his way, and applying his power to them will cause the change to be other than what he intended — and as discussed earlier, this will hurt and tire him. And, he will be unable to help getting some of the spinning transferred to himself during the moment he applies his power.
With the specifics in mind, other problems can be thought of. Can he stop water? No, because a hammer blow at the surface would not transfer to the bulk. Likewise, could he affect a pile of loose sand or stop millions of separate small grains from moving towards him?