We have cycled with and without axies in our tanks. Yes you can cycle with an axie. You've learnt the hard way, what not to do, thru misinformation from that petshop. Am surprised they said that esp. if they keep fish, unless it was someone who doesn't look after the fish tanks.
I think your axie has been thru enuf trauma with the burns, IMO, so I would suggest you place him in a tub of fresh dechlorinated water. We actually did complete daily waterchanges on the tub, my sister did 75% changes every 2 days.
With the tank leave it to cycle and test every few days and as Anna says, no need to do waterchanges on the tank while cycling.
If you choose to cycle the tank with axie in it, you will have to be rigorous about checking the water quality, ie: test the water daily for ammonia and nitrite. If dangerous levels are found, usually 20=30% water changes are needed to ensure it is safe enuf for your axie. Otherwise you'll find the same thing (burns/gasping) and/or illness leading to possible death.
(Message edited by kapo on October 09, 2006)