Help Duplicating Altium PCB Layout With Different Silkscreen IDs
Unfortunately...

I don't think this is supported. Here's a workaround (hack):

1. Break the component links

From the Schematic, 


  Tools -->
  Annotate Schematics... -->
  Reset All -->
  Update Changes List -->
  Accept Changes (Create ECO) -->
  Execute


When Altium reports the number changed, it should equal the number of parts in your schematic. If it won't do it for all parts, find the parts that keep re-annotating to the same ID/designator and manually reset those (double-click the part and click "Reset" next to the "Unique ID" field).

2. Perform a Design Update

Sync the changes over to the layout via the ECO system. 

Altium will warn you that things don't match and allow you to match manually.

3. Fix the pours/tracks

If you have changed any net names, you will need to manually fix the affected pours and tracks. Altium will reset anything that no longer matches a netlist name to "No Net" so be careful to check for this or you will have a very fragile design that will cause you trouble later.