I would add, @Kira893:
You said, "I don’t really care much for my last name, I’ve no contact with any of my dad’s side of my family."
What this says to me is that you consider your own surname to be more your dad's than your own, and that you only have it because the decision that was made when you were born was that you got your dad's surname instead of your mum's. I don't know about the circumstances of your birth or what your parents' relationship was like but I wouldn't be surprised if it never occurred to your mum that she could do anything other than give you your dad's surname. We are so conditioned as a society to believe that men's names are more important than women's, and that women and children must be named according to their relationship to a man, rather than in their own right. And, I suppose, that only little boys can ever hope to grow into their names and "own" them; little girls generally borrow their fathers' names until they get married and take their husband's name.
And yet, here you are, with a three year old daughter who has YOUR name, considering taking a backwards step and perpetuating this cycle for no reason other than that a man thinks his name is more important than yours.
When your daughter grows up, she won't think of the surname she has now as her maternal grandfather's surname. She'll think of it as her mother's surname. Your surname.
If you change it now, there's every chance that she'll look back and think, "Why did you do that? I'd rather have kept the surname I shared with you?"
Or she'll get married and take her husband's surname without even thinking about it, or maybe not get married but give her children her partner's surname, because she'll think, "I'm not that attached to my dad's surname anyway. It isn't even the surname I was born with. Who cares? Names aren't important."
Break the cycle, OP. Be proud of your name that you share with your daughter. Don't change it for a man, especially not three years down the line when it is already part of your daughter's identity.