This exact thing happened to me when I started back in the hobby with a 32 gallon biocube. I bought live rock from an established tank someone was tearing down. Probably quite neglected, but at the time I was blissfully ignorant. It had hair algae on it, but I didn't think what that could mean.
First, that rock was likely saturated in phosphate. The GHA started going berserk in my tank. Like you, I tried dang near everything. I thought I had it tamed down every now and then, but it would then grow back in spades. It was engulfing my coral.
Finally, I'd had enough. I moved everything over to my 210 gallon (every fish, every coral). I took out over 50% of my rock, putting it into a 5 gallon bucket with a PH and heater. The rest I turned upside down, so the algae was on the bottom instead of facing the light. This wasn't intentional, I just wanted a different aquascape. I did a 30% water change.
Next day, my phos had dropped from .22 to .04. Overnight. The water change couldn't account for that big a drop. The drop was a direct result of taking the phos saturated LR out. I'm betting your rock, while it started out clean enough, harbors lots of nutrients, keeping your GHA alive.
I've kept the tank running now as a place to acclimate new fish and stick a couple of frag racks. I mostly ignore it, with the exception of a 10% water change every couple of weeks. I barely scrape the glass.
Of all the GHA I had before, only one clump lived. It's the only one on the top. The rest is completely gone. I've had a gem tang in there for the last 2 weeks (in observation). He hasn't taken a single bite out of the algae. I also have an urchin in there. It ignores the GHA and is, instead, busily stripping the coralline off the back wall. Same for snail and hermit inhabitants. I'm tempted to break off the piece of rock holding the last clump of that stupid algae and be done with it forever. I also stopped running the super bright channel 1 lights, and ran way less white and extra actinic.
So, yeah, not sure how easily it would be for you to do what I did, but it worked. Goodbye, good riddance, and I hope to never see that stupid algae again. It's enough to make someone quit.
Anyway, do a hard reset if you can without starting over. Just putting the algae on the bottom of the structure will do wonders. Temporarily, or even permanently, get rid of some of the LR to see if that helps. I wouldn't have even thought of trying this, but I was darn near ~over~ it. Good luck.
Pics are: way the tank used to be (immediately after hand picking and trimming the algae, but everywhere green became a jungle again in less than a week). Then FTS of it now as frag tank, and last is a closeup of the sole algae left.