With help from @zachary and @jagor I have integrated the Neopixel library with the analogWrite command from the Tinker app.
Gist: https://gist.github.com/emcniece/8526703
However after the first colorWipe() call, the core disconnects from the network (flashes green for a few seconds before returning to breathing cyan) and the app pins all start timing out and showing the 404 error.
I think there’s some info on those issues around here somewhere, will take a look.