Well, the cable arrived. I popped God of War into the PS2 and set everything up. I was quite surprised to see God of War playing on both my TV and on my laptop via the Dazzle, all at the same time.
The trouble is, the colors are "off" on both screens. I tried plugging the cables in several different ways, and each time the colors shifted, but they were still wrong.
I believe what is happening is that the component signals are "mixing" a little bit, which wouldn't be surprising since there are so many Y-cables involved. Anyway, I was basically doing a quick first try. I'm going to try it again (more carefully) in the next few days. It's so close to working.