I found a square 25cm plastic bowl from the Reject shop ($2.50 for a pack of 2) and I already had some spare pieces of mount board handy which would become the rear section of the beauty dish, using the same "slip-on" method from my previous design.
Along with some tape, a glue stick and some aluminium foil I created this:

I have some baking paper that can be attached to the front with small bulldog clips (for added diffusion), though it seems OK without it.
Next time the Wife goes to Spotlight I might see if they have a small off-cut of ripstop or something similar.
I might also try covering the outside in black to help keep all the light within the softbox.
Here are a few quick test shots I took with the new setup (with my reluctant models - aka kids)




Re-edited image:

This will be a little more robust than my earlier design and is also very lightweight.
It gives me something to tinker with until I can get my hands on some "real" light modifiers...