As an update to my 'paste the screen' quest: I ended up just making a function that, when called, individually pastes every relevant onscreen object in the proper Z order.
Thankfully, this has worked everywhere I've tried, including on Cocoon.io Canvas+ with WebGL enabled. Also gives me a more flexibility to omit certain objects conditionally.
tldr: Paster's normal functionality -- pasting objects -- seems to work fine in WebGL.