You can treat him for 30 days in a single QT, if the current fish can move to the 110 without all of the live rock/inverts coming with them.
Can just set it up as a large QT but would need lots and lots of bio-spira as it will be difficult to do water changes sufficient enough to make a difference if ammonia creeps up in an un-cycled tank that large.
But, if you run into issues with the puffer in CP... appetite suppression, bacterial infections, etc.. you won’t have anywhere else to go with him. I would at the very least have a spare 10g you can use in a pinch temporarily if you need to break down and sterilize the current QT.
Edit. Duh, instead of moving the fish to the 110... can move over the live rock/inverts and sterilize the 55g for the puffer. Still would want a spare QT just in case you need to transfer the puffer after 14 days for whatever reason.