In most JW's view, he hates (or at least is SAD) at anyone that is not a JW, not just you. This is why it is stated that he will destroy at Armageddon, anyone who is not a JW. JWs may not admit only JWs will survive, that is clearly the message from the platform.
As you claim to already know JWs do not celebrate birthdays, what did you expect would occur when you asked your mother to help out with anything to do with your son's birthday? While you may not consider a birthday treat as equal to a birthday party, a 'birthday treat' to a JW is a birthday celebration, regardless. If you really want her assistance, don't mention 'Birthday'.
It is not really his birth day as that would be the day he was actually born on, making that date each year after he was born, the anniversary date of his birth. As most JWs do not have any reservations celebrating a wedding anniversary, ask your mother if she would celebrate the anniversary of your children's birth. She will either probably say NO or she will need to do further research or ask the elders if that would be OK.