My rationales…
I assumed that OP uses V6 where (as I have heard) project-cache.lib was included in schematic file so there is no risk if I skip this topic.
What I wanted to say mainly was that you can use file manager that let you copy and rename many files at once. I suppose such functionality should be common in file managers, but have no experience other that one I use so I mentioned its name. It can be useful in some cases. I use it for *.step and *.wrl files also (when I copy 3D models to my library I also rename them to my names).
The other what I also wanted was ask for info where from are the sub-folders if schematic is one sheet. As I never used many sheet schematic I don’t know if they are in the same folder or in sub-folders and this makes it difficult for me to ask unequivocally for what is in sub-folders. If for example OP keeps gerber files there then when copying project there is no need to copy and rename them. He can simply not copy them.
I mentioned that I am still in V5 to not confuse OP if there were some extra files in V6 that I don’t know about.