I have seen lone mums with a baby in a pram do the
Camino Frances, and way back there was an American family came on here and talked about their experience of walking with a child being carried by the father in a child carrying hiking backpack, so it's definitely doable, but I would qualify that by adding in my own opinion it's more manageable on a route with a bit more infrastructure and less challenging sections than the Primitivo, which if you are coming in from the North and want to stay up there, this leaves you with the
Camino Frances and most likely starting from Leon(300-310km), for which there is a daily train from Bilbao to Leon or take a high speed one to Vitoria and change for San Sebastian to Galicia train(s) and there is most likely busses as well.
The Primitivo is a stunning route, and some of the reasons why that is because it has very remote sections with less interference from the outside world, so that is a positive but in your case because you can't guarantee that you will be allowed in Albergues with a child, some will some won't, and this is what I picked up from previous pilgrims who have walked with children and that's even if there is space. You need more options to take private rooms than maybe the Primitivo offers , you do find them on the Primitivo, but the options to take them is less than the
Camino Frances, + on the Frances there is more options to take a taxi to your accommodation if it becomes too much on any day for your daughter and she needs to be in a calm place in a room.
However I'm going to contradict myself here, it might not seem doable until someone actually does it, whatever you do...
Buen Camino