After batting dinos myself, talking to multiple people, and reading a bunch of dino threads, you’ll get the best results running the UV from and back to the display tank. My recommendations are using at least one watt of UV per 3 gallons of system water, plumbed from DT back to DT, at a flow of 1-3 tank volumes per hour. These values are what eradicate dinos and UV manufacturers will give you very different advice. I would also use GAC to help reduce dino toxins. I replace GAC every 2 weeks. Some people eventually can put UV in their sump return after things get better. I ended up plumbing mine permanently to and from my display.
As long as you have good test kits, your nitrates and phosphates numbers are fine, but the general dino guidelines are about 10 ppm nitrates and 0.1 ppm phosphate (I personally wouldn’t mess with either).
The fact that you have hair algae and relatively happy corals is very good. When I had my dinos, even hair algae wouldn’t grow.
Since a good UV is very expensive, you MIGHT be OK monitoring your system and letting it ride out for a while. Keep encouraging algae. Your call based on $$.