The question is about the node, is the node functionality enough in fast synchronization for the functioning of a centralized exchange? Or is it necessary to download a node in full synchronization?
If you're not interested in the history then fast node should be enough, I guess. Fast sync basically just ignores the history and starts full sync from the present time, approximately.
On the other hand if I was to run an exchange I wouldn't want to have to trust the past history checkpoints of fast sync and I'd want to verify the past data myself so I'd run a full node. The cost of running a full node is insignificant compared to all the other costs of running an exchange.