Welcome - the task or path ahead of you is "rocky' and steep. There are no single, cheap "silver bullets" for fixing your filamentous algae problem and high nutrient sink problem, at least not yet. Once a pond becomes nutrient laden (eutrophic) it is very difficult, but not impossible to change the current conditions that feed a heavy thriving FA population that by the way always starts its growth as single filaments attached to the underwater surfaces usually the bottom sediment. As the mat gets too thick for "its own good" it breaks loose and floats. At that point it is usually done growing and absorbing lots of nutrients - essentially useless in the pond waiting to die and recycle its nutrient "load". Since the pond has lots of dissolved nutrients and a lot of bound nutrients in the sediments, I think you should focus on the causative agent by removing or binding a lot of those nutrients feeding the algae and/or possibly channel those nutrients into less problematic nutrient consumers. Again there are no easy fixes.
1. I think you will have to use a combination of management "tools" to be successful long term.
2. Decide what is the main goal? A. Big fish and what type? B. Swimming domestic water? C. Aesthetics? D.General recreation? I know - all of the above as many pondowners desire this, but realistically it is rarely possible without some major efforts - expense in an older nutrient rich pond that has not been previously managed properly.
3. Reconsider tilapia. Bacteria are not cheap and neither are tilapia or the chemicals in amounts needed to keep treating the FA that always wants to grow.
4. Managing a nutrient rich pond to be weed free is not cheap in money time or labor.
5. Carefully read the Thread in the Q&A Archives for filamentous algae management options.
http://forums.pondboss.com/ubbthreads.php?ubb=showflat&Number=92633#Post926336. For the quickest fix, look into having the pond sediments removed by muck suckers. Or consider draining and pushing out sediments of the pond to start over with a clean pond bottom depending on your goals esp if goals are aesthetics with the least effort. Neither method is cheap but it quickly fixes the problem long term without adding lots of chemicals which is also annually costly. Then manage for low nutrient input and proper management of nutrients.
7. Keep removing as much algae as possible. Gradually - long term the nutrient sink will become less. This is time consuming, and slow progress depending on amount of eutrophication. Tilapia help bind nutrients into fish biomass. Some ponds are designed such that one cannot remove enough algae each year to keep pace with the new nutrients that enter the pond.
8. Use phosphorus binders to reduce the amount of available P feeding plants.
9. Use added rooted vegetation to compete against the FA for nutrients. If you have lots of rooted plants consuming nutrients you will have less and less FA as the plants become more abundant. Some people do not like weeds in the pond but they utilize lot of nutrients into another pathway. Southern pond owners often add proper nutrients to create a green water algae bloom to shade growth of FA and rooted plants. This is a slippery slope and if rarely recommended for northern ponds except fish hatchery ponds that are usually drained each winter.
10. Grass carp do not like to eat FA, but at high densities of 30-40/ac they will keep the pond FA free, but expect the water clarity to be around 12".
11. Use more blue dye regularly to keep the light penetration into the water at a minimum - light starving the FA in water deeper than 2-3 ft. Ideally blue due needs to be refreshed monthly (small doses added monthly) to make it optimally effective because the organic blue dye or chemical is always decreasing after it is applied.
12. Change the nutrient ratio of N:P. The current N:P ratio is beneficial to the current specie of FA and not plankton. This is new science and not well understood yet. It will take some experimentation, homework, and professional help to be successful.
Others may have advice. The chemical method is the default method of many advisors. Chemicals can be a relatively "quick fix" but remember the nutrients will continue to accumulate over time usually making the bottom muck / sediments much thicker. Over time accumulating nutrients often result in other types of nuisance algae or plants becoming established such as bluegreen algae scums or floating duckweed or watermeal. IMO there will be no easy solutions and it will probably require several methods to be successful long term. FA loves to grow in water with the nutrient balance that favors it.