Can I do Part III of the Mathematics tripos as a part of my PhD in Cambridge University?
Part III of the natural sciences tripos is essentially a Master's degree: you take courses and exams and write an extended essay. The application process is completely separate to that for DAMPT PhDs (or PhDs in any other of the Cambridge physics departments). 

The simplest option is to do Part III and then apply for PhDs while you are doing it. It's highly unlikely you will have time (and I'm not sure it's even allowed) for you to take Part III courses while registered for a PhD. You could contact someone in Part III admissions to ask them whether you can take courses alongside a PhD, but I suspect the answer would be no.

Personally, I wouldn't place too much weight on the importance of Part III. It has its fearsome reputation for a reason and many people I know who did it, hated it. If you go straight into a PhD, you will learn the relevant knowledge as you go along -- it's unlikely you'll need an in-depth grasp of every part of QFT, general relativity, group theory etc.  Anecdotally, of the other PhDs in my year group (not Cambridge but a similarly reputable cosmology institute), only one out of seven of us has Part III. You easily can succeed in a theoretical physics PhD without it.