My external tools copy fine and still work when I copy one Canvas course to another. However, they do not have any key or secret that is dependent upon or specific to a course. Aside from the key and secret, we do have one LTI we use that has a custom variable in it and we have to update it every time we copy the course.
Perhaps one of these things is what's causing your LTIs to not work?
Another possibility might be the way that you're trying to copy them? Exporting to a file and then importing that file is more likely to break things that copying within Canvas.