idiomatic phrase to describe the fact that the whole family was having lunch

None of those sentences is very idiomatic.

The problem isn't the preposition that should be used in the last part of the sentence, but the fact that the last part of the sentence should actually be towards the start of the sentence instead.

In other words, the following is much more common and natural:

4. Last Friday, the whole family was having lunch at a new eatery.

If you want to emphasize that you were part of the family, then use the possessive pronoun rather than the definite article:

5. Last Friday, our whole family was having lunch at a new eatery.