Thanks to all for the tips here - I just brought my Q200 back to life yesterday. I'd also forgotten that I could remove the burner, but when I finally cleared all the grease off the bolt/nut holding it in place, I took the burner off and used an SOS pad to scrub most of the gunk off. After that, I used some citrus cleaner on the more stubborn gunk. At this point the burner looked clean (albeit discoloured). but the holes were still clogged, so I tried spraying oven cleaner and letting that sit for 20 minutes. It didn't really work, so I found a thick paper clip and used vise-grips to jam the paper clip into all the gunked up holes. I was amazed at how much carbonized stuff was actually INSIDE the burner!
The end result is that the burner is still discoloured along most of the surface, but it looks clean, and best of all it lights all the way around on the first click!