Lanthanum 100% works to lower phosphates. I was just about to post my guide but looks like @BAYMAC and @Darkxerox beat me to it.just be extra careful with the dosing, overdosing can cause severe issues specially if you have tangs! I am also facing the same issue as you.. Unfortunately for me, Lanthanum Chloride didnt work either.. I had a bad case of cyano as well just like you.. I added UV, changed the flow direction and added Rowaphos and my Cyano disappeared.. Though I did turn off the lights for 3-4 days since I didnt have corals, so that might have played a factor too..
Did you test your fresh saltwater for phosphates? If phosphates are high even in your fresh saltwater, then you can rule out Over-feeding as a potential cause and can then look at your RODI or salt to be the potential culprit
I am a big fan of lanthanum, but given that we have certainly identified the problem here - just stop feeding so much, do a bunch of water changes, and see how things go. Your PO4 levels are so high water changes are a very efficient method of removal right now.
Lanthanum is very effective, and while it can definitely be used to drop phosphate levels quickly, the best and safest way to use it is as a slow-drip maintenance approach to keep your po4 level steady at some target.