Hey Mr. Panda. It was about time I learned if these chemicals were actually toxic. I did a couple google searches under 'neem oil toxicity fish' and 'potassium salts of fatty acids toxicity fish'
Sorry I'm too lazy to find these links again and post them, but they were under the first 3 google search results.
Here are the general answers I found:
NEEM OIL - This has been used in aquaponics for a while, but is not used anymore because it has been shown to be moderately toxic to fish. It probably wont kill them outright but it hurts their gills, delays development, overall not good. However, Neem oil is supposed to break down quickly, also it might work on your quarantine tank. If I used Neem oil in your case I would say be careful to 1) Do a large water change, or 2 large water changes before you add fish/invertebrates. 2) Take care of your equipment if you use any nets or anything in the quarantine tank be sure to clean them (probably with soap) before you use it in the main tank. 3) activated carbon might help remove it from the water.
I use neem oil to kill aphids on my pepper plants. These are right next to my 29 gal aquarium. I have shrimp, snails, cory fry, and other fish in there. I haven't had any problems whatsoever since using it, however I am careful not to get any in the tank.
POTASSIUM SALTS OF FATTY ACIDS - This is very promising too. It seems to be not very toxic to fish and apparently degrades very quickly. This is said to be very toxic to aquatic invertebrates though. This might be a good option to use. Again, use the same precautions as the neem oil.
These insecticides are generally safe for plants, humans, and pets and are advertised as a safe indoor pest killer. However...they are toxic to aquatic invertebrates. Be careful with shrimp/snails/crayfish.
Pyrethrins are supposed to kill Springtails, however be careful. Pyrethrin is generally a safe insecticide which is derived from Chrysanthemum plants. However, overusing pyrethrins can aggravate humans and are dangerous to cats. Also pyrethrin compounds are dangerous to all aquatic life in small amounts. I would advise against this.
Generally, in my quick search, I would advise that you try to smush the infestation with your fingers if it is small enough. If it is too big you can try wiping it down with Vinegar for a while, apply this every day. This is a method that I read to kill springtails and it will not hurt anything in your tanks. Too much Vinegar will lower your pH, but you would need a lot, and this generally isn't even a problem, especially with plants.
If none of these work then I would look into Neem oil or Potassium Salts of Fatty acids. I'm not even sure if these will kill springtails, but they are working for me to kill aphids indoors. If you follow safe procedures and keep as little of it out of your main tank, then it should be fine. A few large water changes should remove any insecticides from your quarantine tank. Also, these compounds should degrade on their own naturally as well. Carbon can also help remove it from the water.