Emotions and feelings are often how people express their moral standing. The majority of people won't be able to break down those emotions and feelings into a coherent argument, but at the end of the day I still hold that on such issues involving human life, emotions and feelings which lead to morals are too important to discard.
You say that the argument, "'I feel like the fetus is a baby, so it is', followed by a bunch of conclusions based on the fetus being a baby." To argue this on a moral level, I would ask why they feel this way and what is informing their conclusion. The most likely answer will be religion, and if I had to guess the second most said answer would be personal experiences. That means digging up theological references that support your position.
History of Church and Abortion. I would bring up that the church originally opposed abortion on the basis of sexual promiscuities and never ruled on if the fetus is a human being or not.
As for personal history, that is often used the argument the other way that abortions should be legal for so long, but it is all over the spectrum. To have a discussion on this is impossible. You can either sympathize or try to bring the discussion in another direction. One will never out argue personal experience, and that is my personal experience.
I do think that your position is based on morality. But saying things like we need to just focus on the science, or that how people feel is not important, ignores a vast number of people. Out of 100 people, I doubt more than 5 have really considered this issue in depth. Despite that, I will equally bet that 95 out of 100 have an opinion on the topic. That means there are 90 people who base their opinion of abortion mostly on feelings. If anyone has any actual surveys, more power to you. But anyone I have talked to about this issue just tends to pick a side because they feel that way. You need to get to the reason why they feel that way first, in a way that doesn't make them feel attacked, in order to change their mind.