OP has already told the teacher she will be stopping in to the office on Monday if it did not come out in the wash. If i were the OP, I would follow through with whatever I said I was going to do. I'm not generally one who makes empty threats. DD knows this all too well
If I were OP and this were my child's coat, I would have addressed it at pick-up time when I saw the paint all over her coat. I likely would have gone to the main office then and asked to speak with the art teacher to find out the best methods for removal - since she would be the most likely person to have to deal with such issues on a regular basis and has probably developed a number of strategies. By doing so, the main office would have been made aware of the situation and any fact-finding / offers from administration would have come then.
Ongoing communication with your teachers, school and administration - for good or troublesome issues - is important. If the OP is a regular helper mom, I doubt she'll be viewed as a "problem" parent. As long as she approaches the situation with a calm and level head, explains that she has tried to get the paint out and asks for suggestions on what course of action she should take next -while also mentioning that had she known in advance that this activity was going to take place, she would have provided her DD with different outerwear so as not to ruin her new coat - those in the school will try to help her. It's all in the attitude you approach the situation with. She might also want to be sure to mention how much fun everyone seemed to be having and how much her DD learned from the experience.
FWIW: I deal with a number of "problem" parents over the course of a given year. It never diminishes the education I provide for their children. It comes with the territory.