Multi-body Publish command is disabled in a multiple design body part.
Solution
If you edit into a part from an assembly or Draft file, the Multi-body Publish will be disabled. Close back to the other file, then close the Draft or Assembly. Then open the part itself directly and you should be able to run the Multi-body Publish command.
Notes and References
Hardware/Software Configuration
Platform: INTEL
OS: windows
OS Version: 1064
Product: SOLID_EDGE
Application: PART/SHEETMETAL
Version: V219MP8
Function: MULTI_BODY
Ref: 001-9544909