How to move data from Primary Data File to other data file without destroying replication
SSMS is creating a copy of the table, populating it, dropping the original. hence the error.

I'd create a new filegroup, rather then adding files to the primary filegroup because of how the files are used within a file group

You can move a table "in-situ" to a new filegroup by rebuilding the clustered index using CREATE INDEX .. DROP_EXISTING 

CREATE NONCLUSTERED INDEX PK_Mytable
    ON dbo.Mytable(PKColumn)
    WITH (DROP_EXISTING = ON)
    ON NewFileGroup