@nickpiggott, some weeks ago I did set up pjsip trunks to Faktortel and got it to register without any issues. However, because with pjsip I had issues with anonymous call blocking I reconfigured all my trunks as chan_sip - so unfortunately cannot tell you the exact config I used. As I recall all I had to do was to set the authentication name and the password in pjsip settings and leave everything else with default values. I remember when I tried to manually enter server_uri and client_uri the registration failed. I would delete existing config and re-create a new trunk with the bare minimum and see how it goes.
Also, the Faktortel Support was excellent in identifying some issues I had - my account was created long time ago and I have no access to their new systems unless I upgrade to a business plan and I was told that the new systems provide full support for pjsip trunks. Give them a go.