I tend to agree with you that you don't have to inform him of something that he can easily find out for himself. You are not setting the back-to-school date and time, so I don't think it is your responsibility to inform him. If it's something you plan, then depending on your decree, maybe you are responsible to inform him, or if you get a call from school specifically about your child, then probably. But the time of a school event that anyone who wants to know about can find out, you should not have to communicate to him. That's like saying you have to inform him of the school curriculum, the school uniform policy, the school lunch menu. All of that, along with the school calendar, can be found online. He can look it up.
He should be on the parent email list anyway. He should be getting the same notices you are getting. Maybe check to make sure he is on the list and that's enough. If he is not, it's not really your fault unless it was understood you would complete the information and you left him off.
But my situation is different, so maybe take my advice with a grain of salt. There has never been an official custody agreement between my ex and me. I have had actual custody since birth, but only by default. I have been cooperative with my child's father, making her available to him whenever he asked, but I do not inform him of every little thing she does. And mostly not the big things either. If he called regularly and asked about her, I would tell him just about everything, but he doesn't. Of course, he was not in the same country as us for the first 3 years of her life and he was impossible to reach with any reliability during that time, and it became natural that I make all decisions without him. (And no, he is not in the military. He's from another country that he chose to return to after her birth.)
Now he is claiming I'm keeping her from him because I don't make her call him regularly, but I don't think I have to make her call him if he isn't going to call her. And, he contributes $0 to her upbringing, so I don't feel like I should be hamstrung by his whims. I don't make it another task in my busy day to inform him of everything she is up to, and I don't think I should have to.