There's no particular reason to use a continuous space-filling curve, other than style points. Just expand x in binary and send the odd digits to $f$ and the even digits to $g$. To prove that you get the right measure you should only need to prove it on every box, therefore on every box with dyadic endpoints. These can all be expressed as simple statements about the binary expansion of $f$ and $g$, pull them back, then you're done.
For a continuous space-filling curve, roughly the same thing should work. Approximate $h(x,y)$ with step functions on the various boxes that the nth step of the curve passes through, making closer and closer approximations with each n.
