The question about adding fertilizer is very appropriate. That's potentially throwing gasoline on the fire.
Not sure exactly where (in E TX) this lake is located, but East TX water is notorious to its poor response to fertilization - unless the lake has been limed.
Even if the lake were to respond to such fertilization - with a phytoplankton bloom as the objective - it would take a serious bloom to adequately limit enough sunlight penetration to thwart filamentous algae bottom-growth in such shallow depths; and that much of a bloom could have negative consequences of its own.
Widgeon grass: I normally only see this plant in brackish water environments. Not sure how well it would do in E TX.
Coontail and sago might do well; maybe too well - and potentially trading one problem for another.
It sounds like a very tough situation for devising an effective management strategy.
Does the spring inflow create a constant discharge? Or, does the spring basically keep up with evaporation? If the lake doesn't constantly discharge, does it discharge water during seasonal rainfalls?