I'd remove the bio material from the Fluval. The issue is Nitrate is processed by low-oxygen or anaerobic bacteria. The problem with the Fluval is that it doesn't provide a location (low flow/low oxygen) for this bacteria, so it's great for Ammonia to Nitrite and Nitrite to Nitrate, but does nothing for the resulting Nitrate.
Porous live rock, on the other hand, provides oxygen rich areas for the Ammonia and Nitrite-loving bacteria, in close proximity to inner rock, low oxygen areas that Nitrate reducing bacteria love. Removing the bio-material from your Fluval will force an increase in the oxygen-rich bacteria on your live rock, and hopefully a more efficient processing of the Nitrate.
I haven't used AZ NO3, so I can't comment there. As far as you algae goes, it doesn't cause Nitrates, it actually consumes them for growth. To remove the trapped Nitrates from your system, you must remove the algae (or something else must eat it). But if you like the look of your algae, keep it. It certainly won't add nitrates by being there.
Kevin