It would still be fiddly, but the masking tape is a little more rigid, and with its adhesive, it's a little easier to attach a piece and anchor one end, as you lay the piece down on the surface.  One problem with the little bits of Parafilm was that I would approach the canopy with the piece, held in a pair of tweezers, and touch an edge of the film to its intended spot, like the edge of the framing, say, and where a piece of masking tape would stick, the Parafilm would lift off, and it took several passes with many of the pieces, until I had them laid down where I wanted them.

It occurred to me, I could have packed the canopy with putty again, like I did when I cut it.  I would have had to clean it afterwards, but I have to clean it now, anyway, because there still is some smudging, even with the Parafilm.  Or another second thought that I had was that I could have masked the canopy, while it was still attached to the carrier.  That would have kept it rigid enough to cut the film, I think.

All things to file away for the next time