Battlefield is planet earth in the code geass verse.
All characters have a year of prep.
I give it to Batman, Dr. Who has none of his equipment like he did on his own universe and I have a hard time picturing him making some of the crazy things he does with the technology in Code Geass when in his own universe he had access to things that don't exist in Code Geass verse, that and a year of prep with no knowledge on his opponents wont do him too much good. Batman has shown to be able to use tech from different planets/universes with way less prep time masterfully. We also have no real clue what kind of tech the Code Geass verse has other than what they have shown so somebody has got to give some info on what the Doctor can do and with what items, since I just see Batman busting out a anti-mutant machine or something crazy. This is a guy he could solo the Justice League. Now with Knowledge I may give it to Dr. Who but I still need to see prep. Also, Lelouche really needs to stay out of these battles... In addition, a year of prep is a bit extreme when everyone here save Lelouche can accomplish some of their best feats (At least the ones needed here) in 24 hours prep tops. a week at most for Lelouche. With a year you amy as well say Batman builds a interdimentsional portal to his own verse and make some kind of world ending nuke and blows up everyone else. Since we also have no idea where they even meet I can only assume they randomly come accross each other or something. Thee earlier argument may apply to the Doctor as well but I have no seen what he can do.
Do the characters have access to their equipment standards?
I love Batman, but victory rests with the Doctor Who. I do not see many possibilities Lamperogue Lelouch to take the victory.
The Doctor Who has a time machine at your disposal and with sufficient preparation time, he could find powerful artifacts and create plans and strategies to beat both Batman as Lelouch. He easily could be prepared more quickly than the other two characters.
Now if the Doctor TARDIS has not, Batman would probably win. He is an analytical master strategist and planner able to work under pressure and be able to defeat his enemies with a good time for preparation.
It's a bad idea to confront characters who are one step ahead of others.
Although I hate to admit it, the Doctor would win.