Check your TDS level. High TDS can screw things up.
The answer is simple in this case. Your Free Chlorine (FC) level is too low for your Cyanuric Acid (CYA) level. The amount of active chlorine (hypochlorous acid) that kills algae (and disinfects) is proportional to the FC/CYA ratio. In order to prevent algae growth independent of algae nutrient (phosphate, nitrate) level, you need to maintain a minimum FC/CYA ratio of 7.5% in a manually dosed pool. For a saltwater chlorine generator pool (such as what you describe) the minimum FC/CYA ratio is 5%.
If you don't want to dilute the water to lower your CYA level or to raise the FC level to around 4.5, then you can lower your phosphate level with a phosphate remover. That should let you operate at a lower FC/CYA ratio of around 3% so 2.7 ppm FC with 90 ppm CYA.
It is true that you no matter your longer term plan you need to elevate the chlorine level to get rid of the existing algae and using chlorinating liquid or bleach is the best way to do that without increasing the CYA level (from Trichlor or dichlor) or CH (from Cal-Hypo).