I've not seen this episode, but from the recap in the OP, it appears that this religious sect is a Jehovah's Witness group, or at least heavily based on that group. Some of their core beliefs pertinent to this question are their refusal to accept blood or blood component transfusions based on an interpretation of biblical verses. There is really no way to circumvent this belief by phrasing it as "a mother's blood is the same", so the surgeon simply convinced the mother to break her vow but she internally rationalized it in a successful way.
In reality, cases like this go to an ethics board of the hospital for a determination of the necessity of forcing a transfusion on to a child for life saving therapy. Adult JW members can and have refused transfusions and died as a result, but minors usually have to demonstrate mental competence of the gravity of their decision (if they are old enough, say 10 or so though this is highly dependent on a judge). In this case, of an infant or very young child, it almost always leads to protective custody and forcing the transfusion. Although it is worth noting that even heart surgery can be done without transfusions (though specific techniques, use of cell saver blood reclaiming devices, etc) so it is possible that the surgery could be attempted without a transfusion (of course the episode recap also says they used a doc without a license and bribed him with reinstating his license if he did the surgery, something that would be ABSOLUTELY illegal/unethical across the US and a president couldn't do it anyway).
So, in summary, if this was a JW (or based on JW) case, then the mother did in fact violate her religious beliefs, but simply was given a good rationalization so she could feel comfortable about her decision. The explanation was not scientifically sound (a mothers blood is not the same as the baby's blood) although it has a certain logic to it; if the infant was breast fed then almost every atom in the infants body came from the mother anyway.
Ok, the OP indicated that the posted recap was not very reflective of the episode, so I found a few others (tvmoviefix and inticweb.com that have a bit different description. Here it appears that Baby Grace is already in federal custody, hence the religious groups self-hostage situation. The doc brought in has an artificial blood substitute (none of which actually work in reality and don't in this episode). Thus the mom has to be convinced to donate after the artificial blood failed. If there actually was an effective artificial blood substitute then JW would probably accept it. Most will accept medication and blood expanders like colloid, though there is some individual variability. But most of my post still stands, the mother is convinced to break her belief but is "eased" into her decision by a convincing sounding argument.