Wel, positive ID is essential. otherwise is right on all counts, the Co2 is probably the key, getting the plants going enough to use up the phosphates.
I have gotten hair algae after using pH down which is sodium biphosphate and phosphates were off the chart for three days. I use a rice grain of that now for phosphates addition, that first time I used a teaspoon. oops. But if the plants are going, that gets used up quickly.
The BBA also is cured by CO2, though a tiny squirt of H2O2 can accelerate the process. I've used an eyedropper to add about a ml directly onto the algae and it dissolved before my eyes, then the shrimp moved in to knock it out. I worked on just a few spots, 3 or 4 ml total over 3 or 4 days, never got rid of it all, but it is still receeding.
The nutrafin diffuser is OK for the moneyand fine for small tanks, I use a Plant guild reactor in one tank, a DIY version of it in another. The 29 gallons is right on the border of whether to go pressurized or not and probably too big for the nutrafin diffuser.
It really depends on the light level. I'm Ok with DIY and a low 1.9wpg, though I still have a bit of BBA in there it is sort of attractive on the wood.