Nevermind, I've solved my own problem.
Simply cutting the bottom off the inner bucket and inserting it rather than the whole she-bang allows it to slide much further down inside the outer bucket. It creates a plastic version of the stainless steel false bottom offered by many brewshops.
Now it looks as though I'm only an inch above the nut holding the spigot in place.
It's a good thing I like doing stuff like this and have the tools because I can see why most people couldn't be bother to build one of these things.