You keep the bowl in the tank as well. It makes it easier when you place the new rock in the tank. If you prefer, you can just cut the mushroom and attach it to a new rock in the tank, no need for a bowl. I like to make sure my frags start out on a smaller piece of rock that can be attached to a larger rock when it is time to place it.
Do the cutting out of the tank and place the bowl back into it. Fill the bowl with aquarium water and add a couple drops of iodine. After it soaks for about 5 minutes put the whole bowl back in the water. I haven't had a problem with losing any mushrooms when I use a separate bowl. I have lost some clippings when doing it in the tank itself though.