Several thoughts come to mind;
First I would want to know if there is a ton of phosphate bound up on the surface of the rock that may be feeding the hair algae.
Second, I would suggest setting up a temporary system in that spare tank to keep the fish safe. I would move some of the rock into the spare tank, fill it with saltwater (some or all changed out of the old tank), let that run a couple of days, then move the fish in. This way the fish would be safe, and you can take your time re-establishing the old tank.
Third (to answer my first thought) I would move the rest of the rock into a barrel, fill it with clean saltwater, heater and some powerheads. Let that run for a few days or a week, then test that water for phosphate.
Fourth, if the rock does not seem to be leaching phosphate, then yes I would probably bleach it. If it does seem to be leaching phosphate, then I would suggest either new dry rock (marcorocks.com) OR reading up on the muratic acid treatment to eat the phosphate off of the surface of the old rock.
Last, I'd put the main system back together, seed it with a bit of fresh LR, then let it run as long as you can stand to let it get seeded. (6 months would be nice).
IMO what your LFS guy said sounds good. My only reservation is that it sounds like there is a LOT of certainty in a lot of his statements (at least how you summarized his statements that is). That always makes me leery in this hobby. When someone says they know absolutely for sure what's going on or causing a problem, I start to walk away. If they speak tenatively and or with disclaimers such as "in my experience" or "quite possibly", I keep listening
