As you already know, this is complicated. I understand your concern. I also understand your daughter’s reluctance to seek an evaluation. But early intervention is often the key to preventing a problem from becoming worse (and then more difficult to treat).
The place to start is with the daycare personnel. It’s striking to me that they don’t have the same difficulties. It may be that their approach to him is different from how he is handled at home. Talking with them very specifically about what they do when the little boy is aggressive could be very helpful.
If that isn’t helpful (or isn’t helpful enough), do encourage your daughter to talk to her pediatrician. A pediatrician would not likely diagnose a child with a mental illness. That is the role of a psychiatrist. It is more likely that the doctor will refer your daughter to an expert in children’s behaviors and parenting skills. It may be that she (and you) need to learn some new ways to deal with his outbursts in order to help him.
I wish you well.