If travelling alone it helps. I’ve been asked twice by British airways and Qantas for a letter from dad or birth cert copy.
Yes - take her birth certificate and also a letter from dad too. I travelled to Poland with my daughter and we have different surnames - from London they didn't check anything but on the way back they asked to see proof so I had to show her birth certificate which matches the name on my passport
I was advised when we registered both our girls to take their birth certificates if travelling abroad to prove they are mine as they too have their dads surname 😅