Fungia are rather durable. They are common to shallow reef environments with tons of flow and surges. They even roll around with storms in the wild.
For the fragging to be successful, make sure they are in good flow, good light, and top water quality; they all should be fine. If you are a little scared about doing the fragging, try just breaking it in half and waiting a few weeks for it to heal. After that you can probably frag it again and make smaller pieces, at least that way you can keep a half as your own or as a back up.
I fragged one about a year ago. I broke the skeleton and then used a knife to cut the flesh so it wouldnt tear making a clean cut for the skin to heal. I had a very good success rate and most of the frags were traded. I still have one piece of it left, which is almost as big as the original now, I dont hand or spot feed it or it would have surpassed the original in growth.
The only pieces that didn't make it were little sections of the fungia that were damaged pretty bad in the fragging process. I even had a few little fins off the skeleton that made it and grew. It didn't matter whether or not each frag had a mouth for me, the ones that didn't have a mouth grew a mouth within a month or so of fragging.
Take your time and all should be well.
