Note: I'm pretty much ignoring the big slow particles in the text below, since your argument works without them.
Suppose the door is initially closed, and a particle hits it from the left. The door's mass is negligible compared to that of the particle, so the particle's trajectory is not appreciably changed. However, in order to swing the door open, the particle must have transferred some energy to the door. So the door is not only open but moving, probably at a very high speed given its low mass.
You said the door opens only to 90 degrees, so there must be some object in place that prevents it from opening further than that. Since we're on the molecular scale the collision between the door and this object will be an elastic one, and it will swing back the other way at the same velocity, colliding again with its doorframe and bouncing back, and so on. So after the first particle passes through, the door is not just open but oscillating wildly, and that means it's open most of the time, so it's easy for particles coming from the right to get past it.
Further collisions between particles and the door will tend to increase the door's kinetic energy even more, until the average kinetic energy in the door is equal to the average kinetic energy in each of the particles. (We know this because of the equipartition theorem.) Since the door has such low mass, this means it will be oscillating very fast indeed. So the equilibrium state is one in which there are an equal number of small particles on the right-hand side as on the left, and in which the door is oscillating very rapidly.
You might object to my claim that the door swings back elastically --- you probably intended there to be some damping that makes it eventually return to its closed position. This is possible, but to implement that the door must lose energy over time, which means it has to be connected to a heat bath. This heat bath has to be at a much lower temperature than that of the moving particles. If it wasn't, then energy would flow into the door from the heat bath, causing it to start swinging back and forth even without any collisions from the particles.
If the door is connected to a very low temperature heat bath in this way then there's no reason your setup wouldn't work. The door will preferentially let particles through from the left to the right but not the other way around, and this will result in a pressure gradient that could be used to extract work. But we're not violating the second law here, because adding heat to the low-temperature heat bath is an irreversible process that produces entropy. Essentially, the system is a kind of heat engine that uses the temperature differential between the (hot) moving particles and the (cold) heat bath to do work on the gas.