The only way to control nitrates is to do more frequent water changes. When my girl had fungus in October, I did a water change every other day, at least 50%.
I do not trust the hydrogen peroxide. I use that stuff to kill my own mouth ulcers and they hurt like a broken hand (I would know about that broken hand part...). I wouldn't use hydrogen peroxide.
However, you ordered a new filter? A second one to use? You can't just replace the filters. My girl now has two, both are cycled, one was her old one that was, of course, already cycled, then her second on cycled in the tank, it works well. Two sponge filters. Keep the old filter in! You're familiar with the tank cycling process, it can take like 30 days, and if it is messed up, fungus and all sorts of things, worse things, can occur.
Has anyone else replied to you about that? I've never seen fungus on the face like that.
Oh wait, the salt baths, how long is your axolotl? When I did my girl's salt baths, it was 2 quarts of water (cold, dechlorinated days in advance), and then I added 4 teaspoons of API aquarium salt, let it all dissolve, put an air stone in, hooked it up tot he filter, gave me girl a 10 minute bath. Within the first 3 days, the fungus fell, died completely. She regrew her missing gill stalk by Christmas, again, this happened in October. My girl is about 11 inches long. If yours is bigger, I would do 3 quarts but only 5 teaspoons of salt. The rule that I follow and learned is "2 teaspoons for every quart."