Complement DCFGs and handles
What you want is not necessarily possible. So you have to build grammars
that will make it possible.

Try prefixing both languages by $\{0,1\}^3$ (for example).  That is,
if $L_G$ and $L_H$ are the languages corresponding to your grammars,
consider the languages $\{0,1\}^3$.$L_G$ and $\{0,1\}^3$.$L_H$.

You should be able to build grammars for them, and also prove that
they are complement of each other. You should also be able to show
that since, there are DPDAs recognizing the two languages $L_G$ and
$L_H$, you can build from them DPDAs that recognize your augmented
languages.

If you build the grammars as you should (it is very easy), you will
get your double handle.

The only problem is that, since both handles are on the same side, you
cannot use them for a bicycle, or for a heavy pot.