Little Picardy is a separate bar, it has its own entrance. You can go there without going to a show.

There are multiple bars inside the building that you can only get to when a show is on, and if you have tickets. Personally I don’t think they are worth mapping separately. Not sure about food, but they serve wine and beer by the glass, that alone probably means they require an FHRS rating.

What isn't clear to me (not having been to either) is whether "The Boards" and the pizza restaurant, "Pizza al Teatro”, are the same place or not.

I haven’t been to Little Picardy. It’s possible that you can order pizza there that comes from the same kitchen as Pizza al Teatro, which would explain why they have one FHRS rating.