I don't think you'll find anyone growing tilapia on just duckweed (at least not for long)....

can i ask what "commercial" aquaponics systems you've seen mention that fish loose money?
any successful business will take some marketing..

if you grow show quality koi, you can sell a single fish for hundreds, if not thousands of dollars..
or if you raise trophy quality fish, you can sell to taxidermists for big bucks.. i know a taxidermist that sells the trout he raises for over $100 each - he's also set up aquaponic systems in high schools, the kids grow the fish out (bluegill and yellow perch), and at the end of the year, the fish go into his trophy ponds..

raise a few sturgeon and wait 10 to 15 years to harvest the caviar and make a small bundle...

around here tilapia is cheap (around $5lb).. a backyarder couldn't compete.. but yellow perch sells for $15-$16lb..
striped bass even more..

with a 12 tank setup, you could harvest once a month for specialty sales (restaurants, farmer markets etc)

so these "commercial" ap systems that are loosing money on the fish.. do you think they know the only way to make it work? or just figured out a "niche"?