Is "bestowing anonymity" the right term or expression?
I don't think anonymity can be bestowed. If it is present, it may be preserved or protected.
"Bestowing" implies giving someone something. If you are "giving anonymity" then this will work fine. If the person is already anonymous and you want to keep it that way, "keeping" or "protecting" anonymity may be a better choice.
Generally -- and in the sense that the OP asks about -- anonymity can't be bestowed. It's either present or it isn't. It can, however, be maintained.