A heavy bioload for such a young tank, and the feeding regimen is such that you'll have issues with nitrogen until the rock will be able to catch up with the feeding and waste of the fishes. I am very suspect of the tetra kits, although the presence of ammonia is usually not an issue with these kits unless they are very old. All salts have some level of ammonia in them, even though they claim to be ammonia-free. This ammonia level is very small, but is measurable with a good kit in newly made ASW. Allowing the water to circulate for 12 to 24 hours will allow for both equilibration of the CO2 content with the atmosphere (which brings the carbonate/bicarbonate buffer system to normal values for seawater) and also allows the small amount of ammonia present to gas off.
Several recommendations:
Use ammocarb, it is a mixture of carbon and an ammonia absorbing resin, use the product for about a week, then discard the media and replace it with fresh ammocarb for another week, for a 90 gal tank, use about 1/2 cup, rinse it with RO/DI water, and place it in a media sock in front of a high flow area of the sump. This will adsorb a good part of the ammonia in the water.
Keep running the skimmer on the wet side, this will remove much of the dissolved organics that decompose to form ammonia-->nitrites-->nitrates.
Make a 25% waterchange and allow it to circulate for 24 hours prior to making the change in the tank. It needs to be exactly the same SG, Temp and approx pH that the tank water is. You'll need to remove exactly as much water as you have for the water change replacement. It would be great to use a turkey baster or a power head to blow out any detritus or uneaten food particles in the rock, or to use a siphon hose to siphon out the rock to remove these substances. They decompose to increase your nitrogen product levels. Replace whatever you siphon out with fresh ASW. The water change will not cause the spikes of ammonia, these are caused by either disturbing uneaten food pockets or pockets of detritus while adding clean water. Blasting the rock so that either siphoning for a water change or allowing the skimmer to remove the suspended particles will help correct issues with these substances. If the nitrogen wastes remain high, prepare another water change of 25% and perform it as soon as the ASW (artificial sea water) is ready for use.
The answer to pollution is dilution.
If you use any particulate filters in your system, you need to allow them to capture your particuates during the water change, then once the water clears, remove them and change to new filters media. These must be changes and cleaned every 48 to 72 hours to prevent their decomposition to ammonia and related products. Food and detritus (fish poop) decompose in the presence of bacteria and oxygen in these substrates, so leaving these substances in the filters is just a means of allowing the filters to clean themselves and return these substances to the water column.
Cut back on the feeding. Period. If you lose specimens over the next several months do not replace them (unless it is a herbivore that will keep the tank clear of micro and macro algae). Your fish population density is so high that responsibly feeding these creatures will always present problems with nitrogen and phosphorus products for nuisance algal growth and browning of your stony coral specimens. Although it might be beneficial to octocorals, it will be problematic for algal blooms and desired stony coral coloration as it will stimulate the production and density of ALL ALGAE, including the zooxanthellae in the stony corals, turning them a lovely shade of golden brown. In addition, heavy dense fish populations will speed the demise of your sandbed's functional lifespan, leading to premature failure of the phosphate sink available there.
This is not to say that you cannot maintain a system of such fish density, just that if you intend to do so free of algae, in an oligotropic system relatively free of algal blooms, such densities of fishes will make it quite difficult to continue to have any measure of low maintenance for such a system, and it may be beneficial to consider having a BB system instead in which you can use high flow to maintain suspension of the detrital biomaterials, and allow a strong skimmer to remove these substances from the water column before they have the chance to decompose to their constituent materials.
HTH, I am sure there will be many comments to help you with your current situation.