There isn't one "pro-choice" position - it's a diverse movement, much like the pro-life position is diverse, too. There are people who genuinely believe that abortions should be elective until birth, but as said before, those people aren't the ones who supported banning intact dilation and extraction.
It's well-known (Or so I thought) that when people discuss positions they tend to talk about the dominant ideology within that position, not the fringe minorities.
The reason why he wants to restrict it is because he believes that before 24 weeks, the fetus has no mind and no personhood. After that, he is uncertain and as a result believes that the state interest to regulate abortion, and specifically to limit abortions to medical necessity/health/etc, begins to outweigh the mother's rights to bodily autonomy/privacy.
Why he wants to restrict abortion after the 23rd week or so of pregnancy is irrelevant to the discussion at hand. What's relevant is that, at some point in time, he wants to restrict a woman's ability to obtain an abortion. Therefore, he doesn't believe that the decision to abort should be "between a woman and her doctor" (For if he did, then he would have to say that a woman who wants an abortion her eighth month of pregnancy who has a doctor willing to perform it for her should be allowed to have an abortion), but rather it should be "between a woman and her doctor for so long as the government-- Or more specifically, he-- says it's okay to have one". The latter is the same stance that PL'ers more-or-less adopt. That is, most don't want to ban abortion outright, but rather limit it to those instances in which it's needed. The only difference, therefore, between the PC and PL stances is that PC'ers place the line at which abortions should be restricted to those of need six or so months after the line at which PL'ers argue that abortions should be restricted to those of need (Viability vs. conception).
If abortions were made illegal except in the case of rape, incest, health concerns and severe fetal defects, I'd be PC then, too, because then one's "choice" would be within the boundaries I want it to be in. You see, when you deny people the ability to exercise the choice want, or say that their choice is restricted to the instances you want it to be restricted to, then you're not really all about choice, but rather "pro-what-you're-okay-with"

...And, for the record, you simply can't presuppose that the fetus isn't a person. That's bad form. Really bad form.
The fact that one wants there to be a period of elective abortion at all (<24 weeks) is enough to merit the label "pro-choice".
Arguing that a woman who is raped be allowed to have an abortion instantly makes you PC?
...and you probably shouldn't be arguing strawman positions of a political talk point anyway.
I didn't know I was.