Regularly clean it.
Like every few days.

How often do you use this door? Perhaps a wider, screened cover would do the trick.

We have a similar situation (partially covered stair) and I am thinking about making a raised floor out of pressure-treated lumber that would lay on 2x2's directly on the ground, and maybe with some galvanized screening tacked underneath. I think it would be much less likely to clog between cleanings since the water would drip through freely.

You could also make a simple box frame (like pizza box size or a bit smaller), cover it with screen, and rest that over the drain. Easy to move when you clean. And if it moves' too much, put a brick on it.