Well, I got a few little frags made. They're about half the size I wanted them to be.
I guess I had never looked that closely at the birdsnest before . . . At least I hadn't noticed how the braches fuse when they touch, anyway. When I looked at what I wanted to cut, I discovered it was connected to other pieces: like a big web. I don't think what I did will help the bottom pieces get more light, but at least the frags are sparse enough that nothing will block the light they get.
I just kind of snapped off the braches with the side cutters as suggested, and that worked really well. I only got one little "shard."
They do look quite neat and tidy, so I'm pleased with that.