Sounds like your running a low nutrient/low DO system. Which would help explain why your cheato dies. If your able to keep your system in this fashion without cheato then why add cheato? Cheato IMO surves to purposes. The first being nutrient export ( without nutrients it will die ), the second being for a safe place for pods, but in a low nutrient environment its a futile endeavor, dosing will likely be necessary to keep the cheato alive, which kinda defeats the purpose in purposefully low nutrient a system.
However microalgae tends to thrive more on phosphates than nitrates. So one could keep keep cheato in a low phosphates/ low nitrates system simply by dosing nitrates. Provided a small amount of of phosphates is regularly added to the system as well ( fish food ).
I dont want to comment on the uptake ratio of cheato specifically, because I dont have that info, but typically the uptake ratio for marine plants is much higher on the nitrate side, for example the uptake ratio could be 20:1, thats 20 parts NO3 to 1 part PO4.
So for example your system has 1 part NO3 and 1 part PO4, the cheato is going to quickly remove the NO3 leaving most of the PO4 for micro(nuisance) alga to thrive on.
So by dosing NO3 as counterproductive as it may seem it will actually allow the cheato to remove more of the PO4 more efficiently. With an end result of the cheato outcompeting the micro/nuisance alga for nutrients thus limiting the growth of your nuisance alga.
I havent even touched on iron dosing here which does seem to improve the growth of cheato as well provided the other nutrients are available. The trick is to find a balance between the the three so that none become limited or overdosed.