Have you told the ex, "If you pursue this legally, it will get back to her; I will not tell her, certainly, but parents will find out and then kids will find out that you are going after the school legally -- and it will get back to her. It could have a couple of effects on HER: She likely will at the least have to explain to other kids why you're doing this (and she really can't, she's too young); she will possibly get bullied by other kids if she's seen as 'causing problems for the school'; and the worst long-term consequence is likely to be that she will clam up and stop telling either of us anything about problems she has at school, because she will fear that if she tells either of us, we will react in ways that make things worse for her, or could."
It's sad that he is over-reacting so vehemently and you have it nailed that he's doing so because he's "making up" for being a slacker of a dad for so long (and for still doing only the minimum). Is there a chance that he's also a bit of a blowhard who talks tough but won't really go through with this? I would think that might be the case here -- he might be blustering to appear to be doing something but when it comes down to the cost of a lawyer, might fade away.
This will not be the last time you and he and a school must all have dealings with each other. Once this is done, can you somehow talk to him about how you both should proceed with schools from now on? Are you the "lead parent" who does all the school contact or does he insist on stepping in? Does he get any information from the school sent directly to him? Does he want that? Does he even try to attend positive things involving school, like your daughter's events there (parent-teacher conferences, social events, etc.)? Does he acknowledge that you are the principal or only contact for the school re: your child?
Once this year is over and she changes schools, that might be a good excuse to sit down and say "Let's work out who does what re: school contact and school events." Do make him feel he can attend events and conferences, if you can be that cordial together. The fact he may feel guilty about not being a dad could be an opening to getting him to be more positively involved in her life -- IF you can curb the over-reactions and meddling with what you are already doing with the school.