Hi Chris,
Im so glad you came online to search for answers. The advice given to you at the shop is wrong and fatal to your axie.
Do not use any chemicals to treat your axie. Tea tree oil is even worse!!
In the event an amputation is required, the only person best qualified to make the assessment and perform the surgery is a qualified vet trained in exotics medicine. Please do not attempt to do this yourself. If the wound site looks really bad, i would recommend you bring it to your vet.
Otherwise, what you can do is this. Separate the axie like you have already done. Ensure that the water quality is perfect, especially for the injured axie. Check that you have 0 ammonia and 9 nitrite readings and <60 nitrate readings. Ensure that the water you use during water changes are properly treated with a dechlorinator. Water quality is particularly important now as opportunistic infection like fungus and bacteria would be much more likely at wound sites.
Try to minimise stress in your axie as much as you can. In fact, you can fridge your axolotl during this period. Ensure that the water temp is cool and free from excessive currents. Keep the area dark and undisturbed. If you can provide hiding places would be good as well. At 4 degree celsius and in the dark fridge, your axie would be destressed. Ensure that you do daily water changes with clean dechlorinated water kept in bottles in the fridge to prevent temperature shock. You just need to put the axie in a receptable large enough to fully submerge your axie but allow its limbs and tail to stretch out comfortably. Fill water to just above its back to keep it submerged but not too much such that it floats. By placing your axie in the fridge, you also render the environment less hospitable to pathogens like fungus and bacteria.
Your axie has a amazing power to regenerate. By doing the above over a period of time, assess the wound site regularly. i recommend you take photos to document the progress so you can compare and constrast.
Remember to also continue to offer a variety of nutritious food during this period. Offer wriggly earthworms, bloodworms, blackworms, axie pellets etc. Your axie would be less keen to feed due to the reduced metabolism of being in a colder environment. However, do try feeding throughout. You can also make a mash out of the above ingredients by blending them in a food processor. The resultant mash can be rolled into pea sized portions to feed your axie.
Feel free to message me anytime if you need help. Im a caudata.org addict and i read my mails daily.
Regards