To expand, the hosts should be able to recycle building blocks to some degree to survive for a while, but they need new to grow and eventually even survive. Some water from your display might help with some waterborne fauna, some P and maybe some ammonia, but this might not be what you are looking for with a QT tank by perhaps constantly introducing potential pathogens from the display. Get some food in there and something to eat it and covert some waste.
While ammonium could take care of nitrogen a bit, it won't help with phosphate... dosing a single kind won't do what you think since there are many different types of organic/inorganic/phosphate/phosphorous and food will help with more kinds. Plus, the food will have a real swath of aminos and not just the unknown or limited ones in your dosing... the host and zoox will need the essential ones to function and I only know of one Amino supplement that even lists what is ACTUALLY in there and it is not all of the essentials. If you had a supplement that had all of the essentials, then this might be better if you are sure that the corals get them instead of the waterborne single cell organisms, bacteria, etc that make it a losing math battle for the corals just with surface area. Does your supplement list what you are adding?
Tanks are SO tricky without fish. You just cannot dose N and P like so many people think and have good results - these are not food and do not get building blocks to the corals the way that so many wrongly think. Take a mysis that a fish eats... it has all of the stuff that made the mysis grow and live and turn into a real animal... that gets recycled and released into the tank... the fish keeps about 1/3 and 2/3 move on down the line to get consumed by something else. Good luck.