They’re more shy than I think people estimate, but it sounds like your 125 might be best for the Achilles depending on size when you introduce them. If you didn’t notice the kind of spots and slime associated with parasites like Ich, Flukes, Brooklynella, Velvet, etc. and you didn’t see some fungal or damage to the exterior of the fish, that makes me think bacterial or viral infection. I’ve see a lot of Tangs go down with Septicemia whether the infection was viral or bacterial in origin, they end up dying like many animals, including ourselves from an infection that gets into the bloodstream. If you know anyone who has ever had Diverticulitis or gone septicemic, they can tell you how painful it is. Viral Hemorrhagic Septicemia is something I feel affects Tangs in the way you observed. The majority of life by volume on an ocean reef is bacterias and viruses. Some harmless and some incurable but just how a virus that we can’t cure like chicken pox can come back with a vengeance as the shingles, infection levels, go up and down based on individual resistance and treatment. I would suggest you treat something as beautiful and valuable as an Achilles Tang with as much care as possible. Consider treating it with medication like Nitrofurazone or Kanamycin based treatment, There are some medications that you can put on food and getting them to ingest a medication right away is more effective than most anything else. Some companies make medicated food, there’s one I used to use for Tangs from New Life Spectrum but you’re better off learning how to make medicated food. The Kanamycin should be available at most pet stores or under Seachem as Neo-plex. In this small study of fish farms in Asia, they’re stressing that antibiotics are better used as a preventative treatment without any consultation from a veterinarian or scientist and that actually lowers the overall usage of antibiotics. “Oxytetracycline, ciprofloxacin, and amoxicillin were the most frequently used antibiotics. Most of the antibiotics were reported to be used for both therapeutic and prophylactic purposes”. “Improvement of fish health through better farming practices and changes in feed dealers' and farmers' attitudes towards self-prescription of antibiotic without veterinarian diagnostics may help to reduce the levels of antibiotic usage and thus contribute to mitigating antimicrobial resistance.”
They’re really shy a lot of times until another fish kinda breaks the ice and they seem to really like/be provoked by Yellow Tangs too I’m my experience.