cyanobacteria is actually quite common in tanks. What makes it different is the ability to fix it's own nitrogen, which means in low nitrate situations which would spell the death of other algae or higher plants, cyano can thrive. There are also usually other indications too like low flow/circulation (obviously, because it's a betta tank), and too much light.
The generally recommended course of action is to: manually remove as much as possible and do a large water change, then completely cover the tank so no light can get in - it's important to do this correctly and use thick black bin liners taped to the entire tank, covering all possible route of entry for light, leave the tank completely for several days. The advice is usually for 3, my personal experience was that 3 days was not enough, but 5-6 was more successful. Do not feed, as feeding involves letting light in. The fish will be fine. Then uncover and do another large water change, increase circulation and dose inorganic nitrate to raise the nitrate level.
In practice, I don't know how successful it will be. It took me months to get rid of it, and nothing worked until I increased the circulation by buying a new external filter and putting a homemade spray bar on it that went the length of the tank. Do you have another tank you could move your betta to? Or the faculty to buy one? I genuinely believe that sometimes starting from scratch is easier than treating it, especially where bettas are involved.
At least one other person on here has had the same problem, perhaps posting in the betta section will mean you can get advice from them as to how they solved the problem.
