This is going to sound absolutely crazy, but let the cyanobacteria come in and cover the hair algae. This will take a little while and your tank will look absolutely terrible. Eventually the cyanobacteria will cover the hair algae and kill it by starving it of the light. It may take a few weeks of just leaving the tank to look worse than it ever has. Then siphon out the cyanobacteria and dead hair algae and treat the tank with ChemiClean to kill the cyanobacteria. Obviously before you can say your tank is cured, you have to make sure you removed whatever caused the algae bloom to begin with.
I had a 30 gallon reef and I had the wrong lighting in the sump so the chaeto died and released all the nutrients in the water. I had hair algae covering everything. I eventually had cyano come in and when I had pretty much thrown the towel in, I just let the cyano take over. Then I got sick of it and decided to give it one last try. When I siphoned out the cyano I realized it had killed all the hair algae and then when I used Chemiclean the cyano was done. I don't know if I got lucky, but if you've tried everything else it might be worth a shot. Best of luck!
EDIT: Looking at your picture, your hair algae problem is minor compared to what mine was, so you might be able to still salvage the tank by just doing water changes with ro/di water. You should make a batch of salt water and test the new water before doing the water change to make sure that isn't contributing nitrates or phosphates to your tank.
