You are on the desired path. I think it's a poor wording choice in the game, where the dialog is recycled for both branches. If Frank isn't asked to be the donor, Beth is saying that she did want it to be Frank and that she would try to get Nessa to agree to use Frank for a second child. That makes sense.
Re-using the same line about Beth really pushing for it to be Frank lacks some context if they did ask Frank to be the donor. Would have been better to reword the dialog for that path to have Beth say something like Nessa suggested asking one of her brothers, but Beth convinced her to ask Frank instead.