i recently saw a video, i thought it was linked to from permies but i cant seem to find it right away at the moment
but here is another video in which they grow azolla with the rice -acting sorta as a ground cover:
well that video i saw had a third that grew a little higher, maybe by an inch or two tops than the duckweed, i think it was water chestnut but im not too certain aobut that - might have been watercress
also here is plants for a future's list of plants for an edible pond/bog garden, you could try growing a few of these along with the rice, get soem more diversity going...
We so far have only planted rice and azolla (not really planted but in there). We'd like to try others but have yet to ID them.
Swales could work if you could keep enough water in them to deter the weeds.
No magic ratios I know of...