We can’t verify as you mentioned earlier you don’t tend to test the tank after dosing, the product that as you mentioned will become a nitrogen source for the system that can be utilised by many organisms including coral and bacteria.
If you were to research on the method that you are using right now (carbon dosing) you would understand that the way that it works is by heterotrophic bacteria assimilating nutrients, once you understand this process you can research how heterotrophic bacteria can become limited in growth you may find from that research that they can be limited in the availability of organic carbon, nitrogen and phosphorus.
if one of the three nutrients becomes limited in your system they won’t be able to carry on assimilating nutrients.
If they not able to carry on the assimilation process it will mean that one or two of the nutrients will become more available.
here you may find some basic information on limitations that you research may align with
This article is the way I see nutrient limitations that impact many aquariums in the hobby, understanding limitations could be helpful for better residual management
www.reef2reef.com
Now that we spoke a little on assimilation and limitations you may want to research how Cyanobacteria gets his energy from, you research may show you that Cyanobacteria needs organic carbon and phosphorus in the water column they don’t require a nitrogen source as they get the nitrogen they need from atmospheric nitrogen.
if you were to study limitations in nutrients you could conclude that zero phosphates and zero nitrates (just one needed to create a limitation) will In affect create a abundance in organic carbon this is the point where I believe Cyanobacteria and dinoflagellates take over as they’re main food source becomes available in enough numbers for them to take over.
they will take over mainly because the limitation made the heterotrophic bacteria that is responsible to limit organic carbon dormant.
You will also see many experienced aquarists recommending the increase of the limitations in a way to outcompete Cyanobacteria or dinoflagellates (depending if heterotrophic or autotrophic) to awake the bacteria and reduce the availability of organic carbon that way.
in conclusion there is no evidence that the increase in nitrogen caused your phosphates to go down, it could of been your carbon source also, the phosphates are just on the threshold to become limited and a error in testing could also exist in this low numbers.
I would expect you to just take my word for it although most of what I mentioned to you could easily be checked against information online.
edit:
in addition we don’t know the full content of the product, it has been shown before by respected reefers on here that some products contain organic carbon in addition of other contents, we don’t know the full content of the product to be able to determine if they could cause a nutrient limitation or abundance