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I guess that's true. I'm almost positive they would have done a full on Batman '66 meets Doctor Who if they could have. Pertwee probably would be the best Doctor to meet Adam West Batman now that I'm thinking about it. That was the camp era.

TARDIS wiki says it's the Third Doctor. Probably because of the lab coat. Though based on the 66 date, it should be Hartnell or Troughton. https://tardis.fandom.com/wiki/Cultural_references_to_the_Doctor_Who_universe/2010s

I don't think anyone has brought this up: the TARDIS is in Batman '66 #4 (Digital Chapter 10).