You can pull the ethernet cord out of your PC and plug it back in, you get a DC, log back in and voila no cut scene and you can continue.
Of course you're still screwed if nobody else does this though.
^ This was possible when they first made the changes to MSQ. I'm not sure it's possible anymore.
Though I do not recommend ever running it again, if you're doing mentor roulette either put up with it or leave. For XP purposes the best way is to spam PoTD 51-60 untill you get lvl 50. At lvl 50 you do (in order) lvl 50-60 (this can take around 20 minutes, when you're inside the dungeon you use the Squadron Battle Manual for +15% XP on 2 hours duration), > lvl dungeon > alliance raid > trial. Most of the times your Squadron Battle Manual will still be up for a few minutes after you finish the last trial roulette, you have to do 50-60 first since that takes the longest and you don't want those 20 minutes wasted on your XP buff timer.
^This is during peak time, I've used to do this for quite some time every evening after work, takes around 2 hours though. During the waiting in queue's I either lvl my magikarp on pokemon on my 3ds or play some Switch game.
Now I took a break before HoH was released and I haven't done it since I'm back so I'm not 100% sure how the XP is in there. Maybe that's the new spam to do after getting lvl 60.