I want to call up a mom the day before her son's Bar Mitzvah and tell her to invite my daughter
Why would you want her to be invited?
I feel like her son shouldn't be the only one celebrating. I want him to see that my daughter is on an equal level to him.
It is good to show equality to kids. But, you should not invite your other kids to someone else's party like this.
I just don't want her to be alone, do you think it can cause a lot of pressure for them?
That does not seem very nice to do. You should not meddle in another parent's business in order to get you child invited to a Bar Mitzvah.