Figured I'd write it here, as this is the kind of problem I'd hit multiple times...
Needed to run the following:
sudo dnf install python-devel
fixed the "missing Python.h" error, so that I could run:
sudo pip install -U websocket
fixed the "No module named urllib.parse" error, so that I could run:
sudo pip install -U docker-compose
And then it all worked fine :)