AbigailII: Well a swap rule would work as well i guess:
White makes a move, Black makes a move,
White decides if to swap, the game continues normally after the potential swap.
This would perfectly balance the game,
as long as there is a combination of first moves
at all, that gives Black no advantage.
So basically the question is probably, whether 'White King to the center, Black Queen to the corner' still leaves Black with an advantage, which it probably/hopefully doesnt.
.. interesting stuff ;)

The nice thing about this solution would be,
there were no ugly restrictions to any moves,
just the glitch of the swap.
Thinking of it, this might be the most aesthetic
solution ;)