The bio pellet reactor doesn't feed the cyano. It's localizing the growth of the good, beneficial bacteria within the reactor itself. So, the photosynthetic cyano bacteria now has a competition free environment in your display tank with plenty of light and nutrients to feed on. Suddenly taking a bio pellet reactor offline will usually cause a tank crash since there isn't much, if any, nitrifying bacteria in the rest of the system. Also, testing zero or near zero nitrates and phosphates when there is a nuisance algae and/or bacteria outbreak is to be expected because the nuisance organism is aggressively consuming it. A healthy, well established tank will have plenty of nitrates and phosphates present. Forgetting the bio pellets for a moment, cyano outbreaks are common in new tanks anyway. There isn't large colonies of good, nutrient consuming bacteria established yet. A 5 month old tank is still a very new tank.
I've gotten rid of large, coral killing cyano outbreaks with very little effort in the past. I manually removed as much cyano as possible with a turkey baster and a net, lowered tank temps to 76-77 deg F (where I now keep them permanently), and followed that up with 2 consecutive days of lights out. I also tried a beneficial bacteria product called CyanoClean. I don't think it helped get rid of the outbreak, but it may have helped prevent it from coming back. However, I haven't used it in 6 months and still no sign of cyano. It could be snake oil, who knows. Anyway, I completely got rid of the cyano without a single water change and without any chemical or mechanical filtration in a heavily fed, very nutrient rich system. I think the key is lowering the temp to slow down bacteria growth.
I know you said you wanted to avoid a blackout, but it really is effective. You'll only need to do it a day or two at a time. It's not always open skies and sunshine over the reefs, the coral can handle it just fine. You don't have to cover the tank either. Just leave the lights off. I actually found it better to allow ambient room light to enter the tank during the day so the fish will come out of hiding and eat.