I believe that you should invite the girl and her mother to the party. Especially being that you and her mom are friends and have mutual friends. It would be a horrible thing if the other friends were invited and this lady was not. Because one of them would likely talk about it to her.
As a parent, it can be a very hurtful thing to watch your child be excluded from things. Think of it this way: if the situation were reversed and your child were aggressive toward another child, would you want your child excluded from things? How can a mom go about explaining to a young child that they are being excluded because of a potential to be aggressive?
Just because it is your child's birthday does not make it okay to be insensitive to others. (I have seen people who are adults really making a super-huge deal about their birthdays...it is not pretty.) Children who are toddler age need to be taught about acceptance and kindness. What a wonderful opportunity you would have to teach these things to your child. That is better than any birthday party or any material gift you could give him.
At the party, if you witness this girl being aggressive toward others, then tell the girl and her mom that it is not acceptable. I think if you have some type of activity at the party and other adults are there to help you supervise, then all of the children will have fun.
A very happy birthday to your son and best wishes to you and your family.