It's great to have a cooperative landlord!
Having been a studio rat for most of my adult life, I've had the opportunity to see a number of different techniques applied to the issue of air-handling noise, some more successful than others. Perhaps the most costly was in the new construction of a small sound-stage for motion picture/video work, in which the air-handling system was located in a separate part of the structure which was largely mechanically isolated from the building which housed the studio! Of course, we still shut it down for critical sound recording, but it was pretty damned quiet!
Which brings up a relatively inexpensive addition you may wish to inquire about: A "Hand-Off-Auto" switch is what the HVAC guys called it, but it simply meant a kill-switch for the system that shut down everything without fussing with any thermostats or other settings. The trick, of course, is to remember to reset it back to "Auto" when you're done with critical listening, but the "Off-On" functionality is surprisingly useful.
Another simple technique that seemed to do a lot for the money had to do with the use of isolating couplings between the air-handler and the ductwork. These appeared to me to be made of heavy leather, but I imagine they were likely more involved than that. Breaking that metal-to-metal mechanical link close to the air-handler really seemed to pay off big in terms of noise reduction, though, and I see that this technique is now often used on the new noiser high-efficiency furnace home installations in my area. I think they would be a pretty easy retro-fit if the current system(s) aren't using them.
Finally, of course, is the ductwork itself. Having it lined with deadening batting is very helpful; the really deluxe version uses over-sized ductwork and a lower-velocity fan with quiet diffusers. In the sound-stage, that meant no actual diffusers, just large openings.
Now, I have absolutely no idea whether any of these ideas is practical in your situation, of course, but I thought I'd throw them out while you're still in this phase rather than later!
