My understanding is that the current “conservative” approach to abortion is based on Catholic ideology
Your understanding is wrong. And your understanding of the Catholic position is also wrong, but for different reasons.
How do your poskim, like the Tzitz Eliezer, shtim with Rav Moshe Feinstein’s famous teshuva on abortion clinics?
You seem to have a fundamental misunderstanding of the poskim’s positions. It’s the Tzitz Eliezer who is relatively meikil. Reb Moshe insists on a plain language reading of the Rambam, that abortion IS ALWAYS MURDER, whether it is done by a yid or a goy. The fact that a yid is not executed for it doesn’t make it not murder, it’s just a gezeras hakasuv that this particular kind of murder is not punished by human courts when it is committed by a Jew, but is when it’s committed by a goy.
As I pointed out earlier, this is EXACTLY THE SAME as killing a goy. There is no machlokes at all, that when a goy kills another goy he must be executed, but when a yid kills a goy he is not. Would anyone suggest for even a minute that this means it’s not murder?! Of course it’s murder, and Hashem will punish anyone who does it, but it’s a gezeras hakasuv that human courts are not authorized to do justice in such a case. The same is true for abortion.
The Rambam is also clear that the ONLY circumstances in which it is permitted to kill an unborn baby are the same circumstances in which it is permitted to kill a born baby or an adult — when he is a rodef.