There are a few ways to do it depending on what equip you have.
Cole jaws? Mount the rim with tabs inside , or outside the rim and use tail stock pressure and cut a new tenon, or recess. Reverse and turn.
No Cole jaws? Could turn a piece of wood to fit inside the bowl and just use friction and tail stock pressure to turn a new tenon/recess. Finding center with the tail stock is just a matter of trial and error.
Have a piece of plywood or MDF laying around that's slightly larger than the rim? Mount on a faceplate, turn it round and then cut a recess to match the rim, use tail stock pressure to hold it in place. If you don't have a faceplate, can use a glue block on the plywood/MDF and just use pressure between your chuck/tailstock to turn it.
All the above assume you have enough bottom thickness for a tenon or recess. If the bottom it thin, I'd use a glue block. Glue it on, use one of the above methods to hold the bowl while you true the block.
I've used all the above methods, and they are relatively quick and easy, just remember to take light cuts until you have it secured in your chuck.
There are other more involved methods (like making a donut chuck), but most people have enough scrap laying around to use the above methods.
If you wanted to just thin the bowl by cutting the outside (leaving the inside alone), you could mount it using one of the above methods and slowly work the outside without mounting it in a chuck.
Hope one of those method sounds good to you, or at least provides you with an idea or two so you can fashion something similar.
