Normally, most people would recommend fishless cycling your tank before adding anything. If you are seeing ammonia spikes, there's really no good option for you. All options for removing/detoxing ammonia will disrupt the cycle, but leaving a large amount of ammonia in the tank will be bad for the fish and could kill it. So you have yourself a bit of a quagmire.
If you are doing water changes, and your tank is not done cycling, you'll be removing ammonia from the tank, which is good for the fish. But it'll disrupt the cycle, and you'll just postpone cycling.....aka, it'll happen again, but possibly the ammonia will spike to a lesser extent in the future.
And I don't think you'll just want to continually dose with prime either. It'll disrupt the cycle, but will detoxify the ammonia (again, disrupting the cycle). Plus, I don't think constantly dosing with prime without doing water changes is good for the fish (I might be wrong on that, but I seem to remember hearing something along those lines).
I'm not too knowledgable on the use of Stability. So I can't really comment on that. I always use established media, or do a full fishless cycle using a source of ammonia.
Definitely test for A/Ni/Na, and let us know what they are. I'm still betting you have an ammonia spike. That's a pretty big bio load to put into a tank that is only 4 days old without any established filters or filter media. Even if you are using prime and stability.
Do you have an established quarantine tank you can put the fish in in the meantime? That way, you could properly cycle the tank while the fish sit comphy in another tank.
If it was me, and this was the only tank I had, and couldn't re-home the fish while I finished fishless cycling the tank.....I would do water changes to lower the levels of ammonia/nitrites. I'd monitor it twice per day, and do the appropriate water changes. This will make your cycle take a long time, but would be the best option IMHO. It'll also be quite a bit of work with all of the water changes.