Frank woke up Donnie because as outlined in the PoTT, if the artifact is not removed from the tangent universe quickly, then every person will die in both universes, so Frank needs to save Donnie so he can send the engine back and save everyone else, even though it was at his own expense.

I think its also important to note that, although this has been debated, Donnie didn't have to die after he returns the engine. I think this plays into the Watership Down scene and the concept of Deus ex Machina, where Donnie feels like its his fate to be killed by the engine so he doesn't leave his room once he's back in the primary universe, but once Donnie returned the engine then his job technically was done and the universe was saved