Dye limits sunlight penetration. Which cuts out the bottom of the food chain (reduces phytoplankton amounts). Pond dye doesn't do much for water shallower than 4 feet or so. If you want to have TFS, it will hurt their food source.
The chemicals that you use in the pond are rated for pond use, and I have no qualms about using them. For instance, Cutrine Plus has no time restriction from use to a person swimming in the water. Use it once or twice, add Tilapia. Doing that uses less chemicals than treating the pond every 2 weeks for FA. Dye won't cut back a lot of the FA growth because sunlight can still reach the bottom of the pond in 3'-4' of water unless you REALLY use a lot of dye (think tidy bowl blue).