A 125g tank for a 3" fish is not a small tank in this situation. Tangs don't need a pod presence, so once the tank has completed cycling it is mature enough for a tang, neither of these present unethical situations.
Not talking about this incident in particular, this is referring to the gentlemen that has kept an Achilles in a 90 gallon for some time.
In addition, ich management with this species in a 125 will be a very tall order and is still unlikely to succeed - particularly in a 9 month old tank. Again, I hope I am wrong.
The tank is probably fine in the absence of parasites. Just my opinion, doesn't make it right.
To the op - one option is to give it a shot, but have a backup plan. I take several risks that are not recommended - but do so knowing I have a plan B or even C for that animal should something go awry.
Set up a 20L or longer/larger quarantine tank for the Achilles tang. Start the cycle by "ghost feeding" it. Make sure there is lots of sponge in areas where lots of flow touches it (HOB filters work well) to allow beneficial bacteria to colonize. This way, when you add the sick Achilles to the qt, ammonia won't further his demise. Have cupramine handy, and buy the ammonia badge by seachem because ammonia test kits are rendered useless by many meds including cupramine. The badge is much easier to use as well.
When your fish is breathing heavily, is too infested, or is losing weight, take him out and place in the QT with .25 ppm cupramine (increase to .50 over a few days) to start. It's advisable to dip him in freshwater with matched temp and ph (I use araga milk to increase ph in my dip water) for 5-10 minutes (although humblefish would be better equipped to advise you on this) prior to going in to the QT. This will help provide immediate relief at least to the gill area since copper takes a few days (so does any med) to provide significant relief to ich.
How to identify when it is generally too late (remember it will take 3 days for the parasite to significantly retreat)
1) the fish has a fat belly from nori but his spine and lateral line is showing quite obviously
2) he has stopped eating
3) he is swimming in to power heads for 1/2 or more of the day (indicates gill damage and an oxygen shortage)
4) he is absolutely coated in parasites
5) his behavior changes significantly - such as a hiding fish that used to be out in the open a lot.
Most of the time a fish with one or more of the issues above will not make it another three days. That said, you should try anyway. A few spots is not too alarming. What I've seen dozens of times is a fish has a few spots for a month or so but then they gradually increase. If at the peak time of day that you check on the fish, they don't seem to be reduced in number at any point, he is probably losing the battle. This can be misleading because of the life cycle of ich, but if it seems like week by week (if it makes it that long) the problem is worsening and not improving, pay close attention and be ready for plan B.
At this stage, fish in QT may die because they are heavily infested. The stress of QT may not help matters, either. That said, a losing battle ends typically with defeat. It's a necessary evil. I've brought many fish back but many people that are not proponents of hospital tanks and qt tanks blame these setups for killing fish - when the only time they use them is when the fish is already very sick and was likely going to die anyway. Ammonia is also the number one killer in these circumstances - which is why I suggest that you set it up and get the cycle moving now. You can "seed" your QT with sponge from another tank but that tank ought to be verified ich-free or you'll just contaminate the QT also. But if you plan to medicate anyway, then I would seed because the ammonia concern is far greater.
After recovery you have a choice. Try again and hope that repeated exposure leads to a resistant fish, or rehome the animal to someone that utilizes proper qt procedures and has an ich free tank. This is not easy to do and many do not achieve this.
This is probably not the most ethical way to treat these creatures due to the suffering but it may work. Pray that you don't get velvet as it's far more deadly.
I wish you the best of luck either way. Make sure you post photos!