Update on Mystery snail saved from betta

My mystery snail Lucky had a rough time overcoming the betta attack and I actually thought he was dead at one point, but he pulled through! Immediately after the incident, I bought a 20 gallon and started cycling it. Is it okay to put Lucky in the tank if it’s uncycled, if I keep up on the water quality? His only other option is the hospital tank, which doesn’t have a heater.