I had a bad year long outbreak. I could pull a couple gal worth out every week bad. The effort to get rid of it needs to happen a couple ways. Chaeto to compete with it for nutrients, and animal related removal. A sea hare or bristle tooth tang will pull it off the rocks, then the chaeto and water changed will get the nutrients after the fish/ hare digest it. I found my sea hare to be lazy until I added a 2nd hare, it's almost like it decided it had to compete now vs eating at it leisure. The hare tripled its algae per day with the 2nd one in tank. Also you may have to help your animals out a bit. I found the longer the hair was the less desirable the algae was to the things that ate it, by pulling the worst of the long stuff out by hand, and leaving the shorter sections that were anchored to the rocks made it more desirable to be eaten by the hare. The longer section would go untouched. You are on the right track with straining the food. I also alternated carbon and gfo. With carbon run good quality carbon, not phosphoric acid activated carbon, phosphates just help the algae grow back. So by getting good carbon you don't shoot your self in the foot by putting the phosphate back in that your gfo just pulled out.. I hope my experience helps
A side note do not use caulerpa algae, it is tempting to do so due to its fast growth and high nutrient intake. However when the nutrients become low after the outbreak is over it can and will at some point spawn. This is very bad news. It can wipe out the entire tank. In my own experience it did and killed everything down to the snails wiping out 5 years hard painstaking reef husbandry in an hour.