Itachi would be okay with that. He is generally pretty chill with people hanging around while he cooks or helping him. He and Sasuke cook together too (or Sasuke cooks and Itachi helps here and there) so he can sympathize with how Lightning views cooking if she were to tell him or something. He would probably be willing to cook whatever suited her tastes, though he'd be tempted to try to get her to make dango :B But he wouldn't push it if she didn't like it.

What would Lightning want to cook?