Will Tommen rule as Tommen I of House Baratheon, or Tommen III (?) of House Lannister?
Tommen the 1st.
The current numbering dates to Aegon's Conquest; the kings of the predecessor kingdoms don't count. It was the same in England after the Norman Conquest. Edward Longshanks was Edward I, and never mind all the Saxon Edwards who had come before 1066.