definitely increase the clean up crew. Doesn't sound like a job for a hare, but maybe hermits, an urchin (tuxedo) or turbo snails. Lights out will scale back the initial growth but you want to tackle the problem. Water changes are always a good idea as long as its RODI water. Test that water too to make sure you're not adding anything unwanted during water changes. Your lights could also be throwing off too much "green" or "red" LED. Check those if you can (I have Hydras to adjust easily)
IMO I would definitely try a second phosphate and nitrate test just to confirm results. Either that or take to a LFS to test. I had a bad test for months telling me my nitrates were fine, only to find out on a whim the testing results were bad (bought test off Amazon with no exp date, so who only knows how long it sat on a shelf) Now, I have 2 kits for both parameters that I double check every few months or so to make sure the main test kit is valid (different company test). I'm a big fan of Hanna's Phosphate and Nyos for Nitrate.
Sometimes I wonder if I'm a captive to these testing kits...
I've learned its usually never one thing for most issues, so the more you can test, the better.