The immediate reaction most people have when they hear that a dog has bitten a child (especially if that bite is to the face) is that it should be PTS , the thinking being that if a dog has turned on you once it can never be trusted.
I was bitten on the face by my uncle's dog when I was 9, broke the skin very slightly, but nothing that required stitches and no scarring, so perhaps the same degree of injury as OPs daughter. My cousin was only a toddler at the time, so without the backstory it would sound obvious that having him PTS was the only option, he clearly couldn't be trusted around children.
There was a reason behind the bite though, he was 'protecting' my cousin from me. She was running (toddling really) away from me, while I crawled after her, 'chasing' her. She was squealing and shouting no as I teased that I was going to catch her, and the next thing I know the dog has bitten my jaw/side of my face. I remember he actually held me there for a few seconds, not really biting down per say, but with his teeth still around my jaw keeping me still.
There is absolutely no doubt in my mind that he thought he was protecting his family, I had unknowingly 'provoked' him, or at least provoked his protective instinct. The baby was squealing, I was chasing her, I'm sure to him it sounded like she was distressed. He was a big dog, if he really wanted to hurt me he could have done so very easily, I think he was trying to stop me more than trying to hurt me.
In the end the dog had to be rehomed (full history disclosed), because my uncle was 99% sure that he would never have hurt the baby, but that 1% of doubt kept him up at night. The whole family was heartbroken, but it was the only option for them.
Sometimes the situation just isn't black & white unfortunately. In the OPs case, she doesn't know what really happened in the room, so staying away from the dog seems like the only option.