The stone does not disappear after use.
It must be on your person to use it for the transmute.
I personally keep mine on me, but thats because i dont do arc transmutes, I do heart of fire transmutes, so the timer is up every 10 mins not 2 days.
For the purpose of arc transmutes, put the stone in your bank until the timer is up. And just pull it out for the transmute and put it back when done.
That should be easiest for you.