A few things that I think might help:
"Sorry, our venue is small, so we only can invite # of people. We only can include immediate family/cousins/whatever your cut-off is"
It's normal to set up additional activities for children to keep them occupied, but you can apply the same principle with any guests that worry you. One of my colleagues had a "Mom No-er" at her wedding. She chose a good friend to hang out with her mother and say "no" to all of the ridiculous requests that she knew her mother would pose. At my own wedding, we have a very dear friend who tends to drink too much when he is stressed. Given that his ex-girlfriend will be there with her new husband, we are going to make sure that other people know to keep him positively occupied. Following the first bullet means you will end up with one or two people whom you would rather not attend, but it's all about making sure that they are positively occupied and not raining on your parade.
One thing that was super clear from your post is that your dad loves you - a lot. He screwed up, but it sounds like an innocent mistake stemming from excitement. As much as the resulting situation must be super stressful, make sure that you recognize how lucky you are to have a dad who is bursting with happiness about his little girl (or boy) getting married. I am sure you are annoyed right now, but it sounds like you have a really amazing, loving father. In the midst of this stress, make sure that he knows that you know that!