FCM and Externally Managed Dimensions are two separate features.
If your dimension is being created by FCM and you want to change the name of the dimension itself while keeping the same ID, I think you'd need to go into ./data/state/fcm.dvals.xml[.gz] and ./data/state/fcm.conflicts.xml[.gz] and rename there, as well as change it in your FCM data feed (where you pass in the dimension schema). If you are actually talking about EMTs, I'm not sure - possibly the same approach. Certainly just renaming them in the data feed/schema would assign a new ID as the logic would naturally assume this was a new dimension (usually there is a separate display label attribute to support business users changing the label while the name is used as a system ID).
Adding to what Michael already said.
- What is the purpose of keeping the Dimension IDs consistent? Are they being used in Business Rules (for triggering or for boost/bury)?
- How often are the dimensions renamed in the external system?
- Is there any other unique identifier that can be used as the dimension spec?
You can have the dimension name, as set in the external system, to be used as the Dimension's display name. However, you should use an attribute (such as intention ID), or a combination of attributes, from the external system as the Dimension spec. As long as this Dimension spec remains consistent, the ID assigned by FCM (PCI) will not change.
Hope this helps.