Putting together a test enclosure to see how my estimates work out. It Is looking though a 5cm height at the back and 3cm at the front will in fact leave enough space inside for the Pi and hat. I may add 1cm to each height for comfort.
Just need to rewire my switches, adjust my code for the new audio interface and it should be ready to test. Once testing is complete it'll be rehomed in a much better case.