Hi A. -
I know a little about what you are going through. I have seven stepchildren (13-21) and we have a daughter together (2 1/2). We have been married over 5 yrs, so they were about the same age as yours when I came on the scene.
I know it can get very hectic and chaotic. There are times you do want to run away.
I realize that you are starting a business, and time needs to be spent towards it. Have you talked to your boyfriend about taking on a partner/employee who could assist with the weekend runs? Yes, it would dip into your profits, but the peace of mind it provides might be a worthwhile investment.
In regard to the ex. You are both right. The time spent at your home is for your boyfriend to spend with his daughter. It is not fair for her to not see him during this time. But, you are right in that she cannot say what happens in your home.
First, does your boyfriend have a "Right of First Refusal" clause in his custody order? If he does, then that should be followed if he is working. If not, then there is really nothing that she can say. If I were you, I would tell her that she is right, this time is for his daughter to spend time with him and she is welcome to come and pick her up if it is a problem for the daughter to stay with you and the rest of the family. If she is truly wanting a "weekend babysitter" as you say, then she will huff and puff a little, but not pick her up.
Maybe during this start up time, you could ask to change your visitation temporarily and have the daughter sometime during the week, when it is not as hectic.
Good luck!!