They don’t need anything for the Bar Matzah, just a birthday gift would suffice.

Yes. Why wouldn't you?

2 months apart AND you see her in person both times? Of course. The birthday gift can be a 20 dollar item but you'd really not give her a present , when attending her birthday celebration, because MONTHS later she is going to have her bat mitzvah?

Of course not! Go with one big gift for the more important one and just give a card or nothing for the other.

I would just give a birthday gift.

You could split your budget in half to do one “half-priced” gift per event.

Yes

Of course you cheap a$$.

Yes, cheap-o. Those are two separate, milestone events.

Yes; these are two very different celebrations.

If her birthday was in a different month, it wouldn’t cross your mind not to give her a gift.

Yes. I am not Jewish but I don't understand why this is even a question. The bat mitzvah is a significant religious event. Of course she should receive a separate gift for it.

Yes

I would.

Yes

The birthday and the bat mitzvah are about 2 months apart, and we will see her in person for both occasions.