In case it helps someone figure it out, I managed to access the trainer again yesterday (I think what happened was that I left it open for 10-20 minutes and when I came back the login screen loaded) despite not being able to use it for the past few days, and now it's gone again today. Of note is the fact that the problems with the trainer started when my Artix launcher for DF also started having trouble loading the game i.e. stuck on the blank red screen. Hopefully that helps someone diagnose what the problem is.
Edit: Also if you right click then select 'Play' on both of them (Gray screen for Trainer, red screen for an Artix launcher crash) you get the 'Error loading Game Engine' message.