I think its interesting that Esme chose to side with Bella and not Edward over the pregnancy. In the earlier books you see that Esme is willing to do anything to make Edward happy and also anything to keep her family safe. It seems like that would have lead her to want to save Bella for Edward and protect her family from the unknown fetus that could be a threat to her family (should it be as horrible as an immortal child...).
Perhaps her not going that route is evidence of her accepting Bella has her daughter, and now it is Bella's happiness that she is fighting for...
Even with Bella, who I think Esme did love as a daughter, I think she always had a soft spot for Edward and, if she had a favorite, it would always be Edward. But, in this particular situation, I think she truly agreed with Bella and understood first-hand how difficult it would be for Bella to lose a child. Esme was so upset when she lost her own little boy that she tried to kill herself. Maybe she was worried that if Edward got his way and aborted the baby, Bella would be so sad that she would try to kill herself, and then Edward would have lost her regardless. I don't think this would have happened, but if Esme was basing her decision on her own experiences, it makes sense that she would think it was a possibility. Esme showed that she loved Renesmee and treated her like a beloved granddaughter from day one. For Esme, Renesmee was never "the fetus," but always her grandchild.
As far as the fetus being an immortal child, I don't think that is possible. I think immortal children were human children who were transformed into vampires as children. Renesmee was half human and half vampire, not an immortal child. Esme, like the others, was probably scared that the legends of the incubus were true and Bella might die during the pregnancy. But, her understanding of how difficult it would be for Bella to lose the baby would make it so she would never side with Edward on aborting it, even if she thought it might kill Bella.