I've seen the chemicals work, I've also seen grass carp do the trick.

hmmm..."don't want to add invasive species", but then suggest to add chemicals... not sure about that suggestion. chemicals are a great tool but I would use them as the last option. Adding a grass carp by itself would not hurt anything. Even just adding two to a would not do anything to the population. Once a grass carp gets so big though they do not eat as much. Do some research first though...

There is another option cheaper than Sonar or Avast. You can spray with Reward (Diquat) there is also a store brand version sold by Tractor Supply. The Reward must be applied as a contact spray over the Duck Weed. This product will also kill the gras around the banks. Triploid Grass Carp are usually not effective against duck weed and water meal.