There's no need to overcomplicate it.
All thats needed is to understand two simple and profoundly liberating concepts:
Q. What does blood symbolize in the Bible?
A. It symbolizes the soul, the life, of the creature.
Q. In the Bible, the prohibition on blood always involved the death of a soul. Does a blood transfusion involve the death of a soul?
A. No, a blood transfusion does not involve the death of the soul giving the blood. There is no loss of life by the giver of blood, and so it has no symbolic value in Gods eyes. Therefore a blood transfusion is scriptually permissible.