The first sentence:
If I hadn't got up early, I wouldn't have been tired.
Is indeed third conditional - it talks about the past and raises an alternative unreal scenario and what follows from it: In the real past, I got up early and (then) I was tired. In the unreal past, I hadn't got up, and then...
The second is a mixture of second and third conditionals...
*If I didn't have to get up early, I wouldn't have been tired
A correct second conditonal would be "If I didn't have to get up early, I wouldn't be tired". This suggests that the scenario hasn't occurred yet; maybe the narrator thinks about having to get up early tomorrow and the likely result of being tired afterward. But it's not a done deal - perhaps he/she may get up late and not be tired after all.
The third sentence is a correct third conditional... if you want to emphasize the obligation using "had to" then this is a precise way of expressing it!
You may feel uncomfortable with the "hadn't had to"; many people "just don't talk like that" - it's a relatively complex construction. Except for third conditionals (which are uncommon) Past perfect is so rarely required that people sometimes avoid using it, or are puzzled when it occurs. But this is the correct way to use "have to" in past perfect and in third conditionals.