source code should not go in a header and that header files should be
kept to a minimum.
This is a popular assertion, and while it may generally not be bad advice, you should not draw absolute conclusions from it. Sometimes headers should be minimal and not include definitions. Sometimes the opposite is true. There are reasons why you would do one or the other, but "people say" is not one of them.
Consider the C++ Standard Library. Better yet, consider Boost. Some prestigious C++ experts have said that Boost is the most well-designed C++ library in history. But if you look at the libraries you'll see that they are basically just giant header files for the most part. How does this reconcile with what "they" say?
The point is this: you must understand the reasons why certain files are designed the way they are, and make up your own mind about what is Right and Wrong for each situation.
Should I copy it all to a .cpp file and create a new .h that just
declares the functions, structures etc?
I would say probably not. This sounds like a recipe for a maintenance nightmare to me. It would be my first instinct to use the 3rd party library they way the library's author intended it to be used. That way you won't be off the support grid, and you won't introduce a new set of complications that you will be completely on your own to figure out. Unless you have a specific, provable reason to alter the architecture of the library, don't. "They say" isn't a good enough reason for me.