Original files are never touched in LR. The edits you make are stored in the catalog file and in a XMP file if you have that option selected.
If you want copies of files do that with the file manager for the OS you are using and start a new catalog for your portfolio.
The originals are on a network drive (my archive) and it is slow to access them. ...
For practicality I'd like to grab copies of the JPG and CR2 files so that I can tweak them for my portfolio. And it makes sense to do that on a local drive rather than over the network.
If your network is performing properly, I strenuously disagree with this. Properly performing networks will not exhibit slow access. Many people have photos on network drives without speed issues. You DO NOT need to make a copy of the photo on your local drive to achieve speed improvements. So I suggest your network is not performing properly.
What I would like to do therefore is not to create virtual copies but to have a full set of copies of the files I want to work on so that none of my edits affect the originals (they will end up becoming TIFF files).
Obviously, if you want to do that, you are going to perform extra work (creating of this actual copy using your operating system and importing it into LR) to avoid one of the built-in features of Lightroom, virtual copies, which takes much less effort on your part, and provides every benefit you seem to need in this case (except the speed you want). The choice is yours, but I really suggest you investigate this network speed issue first.