Second thought: Am I a terrible person for rather enjoying the Doctor when he's in "I am The Doctor, destroyer of worlds" mode? Particularly with 12.
If so, we can be terrible persons together. I don't want him to really destroy worlds but I like it when he's threatening and dangerous like that.
So we are stuck in a room with a stasis chamber, a human who cannot be killed and an alien who when he dies can regenerate and no one can figure out how to save Clara? The least the writer could have done was give us reasons why none of those would have worked.
On the one hand, it's no worse than Amy and Rory, where there's plenty of ways the Doctor could save them, but doesn't apparently. Which is probably the one problem with this "part-time companion" thing Moffat does. When it comes time for them to leave, the only option is apparently something completely convoluted and doesn't hold up to scrutiny.
But it was explained. Once Clara had taken the chronolock she couldn't give it to anyone else. Ashildr explained that when Riggsy said he'd take it back. She wouldn't have been safe in the stasis chamber, either. It was emphasized a couple of times in the episode that you can't run from it and that it will find you wherever you are.
I also think that one of the points the episode was making was that you can't always cheat yourself out of a situation. The chronolock is a metaphor for facing certain death, a bit like the Kobayashi Maru in Trek, only it really kills you.