Common opinion in my area has been that "plain ol' river carp" will eat duckweed, but complete eradication by "natural methods" is pretty much impossible. If you put in enough fish to eat it all, when they have eaten it all they'll starve and the stink from the dead fish might be worse than the weed. (Using goldfish or koi is frowned on in some areas for open ponds where you can't strictly control possible "escape" breeding.)
Initial growth seems to be mainly in shallow water, and adding some taller "water weeds" like cattails or one of the taller "marsh grasses" sometimes slows the duckweed by shading the shallows. Shoreline trees might also help a little, but I think you've indicated you've got plenty of them already.
Increasing the in/out water flow to increase circulation is sometimes claimed to help, but of course if you don't have enough "in" increasing outflow won't favor having much of a pond.
It's likely, if duckweed is more of a problem than usual (although "usual" is bad enough), that you're getting some "excess fertilizing" from feedlot or cropland runoff, but that's likely to be beyond any controls you can apply beyond encouraging your local ag agent to point out to the people upstream that they're spending a lot of money on all that excess phosphate they dump on their fields. (Most "real farmers" here have soil sample testers and computers to work the cost/benefit trades very closely for their cropland, and don't produce much runoff, but you can't keep the pigs and cows from dumpin' when they feel the urge.)