I dont like the fact that her mother and me have different views and it must be difficult for her to know which way to turn
It's going to be a difficult few years for all three of you. The mother is bringing up her daughter in the way she believes pleases God the most, however wrong it is. You want what's best for your daughter too. As for your daughter, the situation is confusing and potentially distressing.
At such an early age, she is not ready for the complexities of doctrine. Even though you are right that Rutherford's ban on birthdays cannot really be supported by scripture it is better to compromise for the sake of your daughter.
Explain that you love her and will still be giving her presents at that time of year, but not necessarily on the exact day. Tell her they are not birthday presents so that she can accept them without feeling torn between both parents. Explain to the mother in advance of your desire to find a mutually acceptable solution.
When the child is older you will find she is much better able to see through the Org. and its false teachings. She will also see that you always loved her and provided gifts despite the religious obstacles.
It's not an easy thing to do, since you are actually in the right, but the child should come first and, at that age, they are just looking for the love and approval of both parents.