Lets assume you have two real constant sets:
real,1
real,2
real set 1 you say is being used for both, but you want real,1 for the original mesh and real,2 for the copied mesh. All you need to do is copy the mesh as you have done but then:
1) select the newly created elements after the mshcopy command and then
2) go to Main Menu>Preprocessor>Modeling>Move / Modify>Modify Attrib
or
issue the emodif command
3) and modify the real constant set to real set 2 for the copied elements
your command syntax will be something like:
emodif,all,real,2
4) reselct everything and the job is done!
-- drej --