Collector improved.

The first attempt was too wrinkly, and the tape used to pull the mylar tight was slipping a bit in the heat, so instead of pulling out points on the surface with the gaffer have instead lined the peripheries with bike spokes and flexible wire, pulling out the entire edge uniformly with cable ties.
I've also added two extra bits of wood lengthways to support the mesh, as it was bowling a bit in the middle.

This has the effect of making the thing pretty much perfect.
I'm actually amazed how accurate the shape is, as you can see here; the focus is about 5mm wide, which is I imagine what you'd get from a professionally machine produced reflector.
Just to reiterate that this has so far cost me about €10 in materials and can be whacked together in a couple of hours with a jigsaw and a drill.

I tested the balance and it's about 50 grams heavier on one corner, so two medium size bolts should bring it back to perfect, thereby having the thing require practically zero force to rotate.