I woke up this morning to an email from our hosting company letting me know that the recent issues with the server since the last management interface update seemed to be connected to the server installation (i.e. the actual physical servers software, not the Minecraft install).
As the problem reoccured at around midnight last night they decided to just reinstall everything, but leave the server down for me to check over before starting it back up, so it has been down since then. I have now checked things over and everything seems fine (all our files were kept safe during the reinstall) so the server is now back up.
Hopefully we're now back to being stable again :)