The Doc would need to use the dial code every time she wants to hide her caller ID. Using the dial code is NOT a one-time fix. It must be entered by the Doc with each and every call.
That said, it may be possible to set up a 2nd trunk from Flowroute and not specify an outbound caller ID on the trunk, but this is a conversation to have with Flowroute.
As jhogue1975 suggests you could route calls from the Doc's extension out the 2nd trunk. Or you could have the Doc enter a prefix like "9" when she wants to hide her caller ID, and omit the prefix when she does want to show the caller ID. The prefix would be used by your outbound rules to determine whether the call goes out the first trunk or the second trunk.