All the detail is there, but the overview is that I didn't have write permissions for the correct fonts folder - and DOMPDF (et al) couldn't write the caching file.
Once the cache file (dompdf_font_family_cache.php) was created, and the fonts were copied into the right directory, it all started working.
This was slightly harder, under the direction of BrianS, I fixed positioned the header - which made it happen on every page (including page 1), and then I absolutely positioned the first page header over the top of it. (or at least, I tried.)
I did get it working - after trying all the expected CSS fixes for layered elements, I tried z-index:-1 on the fixed element. That did the trick.
Thank you BrianS for all the support, help and insight. Appreciate it lots.