The dome acts as a coarse filter, straining out debris and letting the water through ... but it's substantially undersized compared to the area it's draining. You need a dome strainer that is more like 16" diameter. You will probably need to rig something up since the pipe is going to be much smaller. As a temporary measure, you could create a shallow "fence"-like structure surrounding the drain (2' dia. x 6" ht.) made of silt fence material or something like that ... anything to filter out the larger debris.

A better pic after the storm and clearing it out.

Thanks Yardvaark, I was kind of thinking that when I was out there today. Crazy thing is it used to be just a flat grate that would instantly get clogged. The dome already improved things a lot. I'll look into that.