The only thing I can find about Bruce Lee and Batman was that he refused to film a scene where Kato would be defeated by Robin. They turned it into a stalemate. Of course the truth would be that Bruce Lee would have been so efficient that it would have been over in seconds if it were a real fight. But of course he had to show some flashy moves just because there was time to fill.
An additional two-parter, to air as part of the Batman series, was shot in which The Green Hornet and Kato teamed up with Batman and Robin. Batman was the more popular of the two shows—and the fight took place on their show after all—so the original script had Batman and Robin winning the fight. Bruce wouldn't have it. "Burt Ward was absolutely petrified when he was going to work with him, and he didn't want to work with him. And Bruce got the script, and in the original script he lost to Robin. Well, that didn't go over too good with Bruce," said Van Williams in a documentary. "He walked off the show. He said, 'I'm not going to do that.' He said, 'There's no way that anyone would believe I go in there and fight Robin and lose.'" Bruce wasn't alone in his incredulity. "Even before I knew about kung fu and Bruce Lee, I was laughing when it was Robin versus Kato," said Meyers of the matchup. "It was so obvious that Robin would have been a smudge on the carpet within five seconds."
The ante-upping continued off set. At a press conference discussing the pairing of Batman and Green Hornet casts, Bruce's first words were in Chinese. But he didn't need language to set him apart. The Green Hornet himself Van Williams averred of him, "I made the mistake of sneezing when I was too close to him, and I ended up flat on the floor. He's very fast. Very fast." When Adam West asked Williams if he was faster than Robin, the Green Hornet actor replied. "Faster than Robin. Faster than a speeding bullet. Gosh darn, that's the wrong one...." As a compromise, the scene was rewritten to have the fight end in a draw.
On set, the story goes, Bruce went icily silent. In his book Boy Wonder: My Life in Tights, Ward explained that leading up to the fight scene Lee didn't say a word, he just stared at him and acted angry all day. When the camera rolled for Kato and Robin's face-off, he stalked Ward until Ward backed away, saying that it was only a TV show. As Bruce continued to close in, someone offstage squawked like a chicken, and Bruce broke into gales of laughter. "Lucky it is a TV show," he cracked.
This was the scene:
Of course with all the newer versions of Robin, he's a world class martial artist who can even take on Deathstroke.