The peroxide and vinegar treatment makes me nervous. I think they could potentially wash into the soil and mess w the ph. The safe way I learned to get rid of algae is a spray bottle of rubbing alcohol. When I lived in a San Francisco, the algae problem was constant because of the fog. Everyone in BSSF used rubbing alcohol, myself included. Just spray it on. Safe on foliage, bark, pots, you name it. I didn't grow Bougainvillia, but it worked on pine, maples, junipers and the other usually suspects.
When alcohol dries it disrupts and breaks the cell wall of the algae ( does the same to bacteria, hence why we use it as a disinfectant.) and kills it. A few days later, poof, algae is gone. Sometimes a white ghost of any algae growing on the outside of pots, but otherwise it worked like a charm and had no discernible drawbacks.
Test it out first if you are nervous, but I would buy the big bottle and attach a sprayer and do it about every 3 months.