Yikes..yes, I managed to do it.
Here's a tip for timing your initial double jump right:
Let yourself get killed, then press 'R' (to re-su-taa-to), then IMMEDIATELY begin running left, and just just a tiny tiny second before you hit the very edge of the platform. If you find that you touch the moving platform on the way UP (triggering the spikes) jump just a tiny bit earlier.