Being it's a 10 gallon tank, I would replace all of the water, not 20%, to get rid of rhe ammonia. Get the tank as low as it'll go so the fish stays submerged. Put your corals and their rocks in a bucket of heated saltwater. You can also remove some rock. A 10 gallon is VERY small and 20 pounds of rock is taking up too much space. Cut that in half. Or better yet, use a 20 gallon and use the 10 as your quarantine tank. Petco has 20 gallon tanks in sale right now for $20.
I have used copper on Valentini Puffers just fine (and 3 other small toby puffers as well). You just have to do it right.
But getting rid of that ammonia is a high priority. If it doesn't outright kill the fish, it'll definitely make it sick and disease prone. Also, if you buy live rock, try to get cured rock in the future.
What type of filter do you have on this tank? Because it was definitely way over stocked. The smaller the tank, the more on top of it you have to be because things are so concentrated that a small shift in anything can have devastating affects very quickly.