Yes, it is a ludicrous scenario, but OP's entire worldview is equally as ludicrous, because it allows for such scenarios("If I refuse to allow it to be used to support another life, that is my choice."). In any case, no special rules or consideration are generally needed for normal scenarios, so you are often looking at less common or ludicrous seeming scenarios when probing the limits of someone's viewpoint.
I'm not positive that allowing no questions asked abortions 1 day before the due date will minimize human death and suffering. Instead, I think it would provide an avenue to take for people who have, basically, cold feet about being a parent.
In fact, I have to wonder if OP thinks they have the right to abandon their already-birthed baby in the wilderness. After all, what right does that individual have to force another person to take care of it? I'm an individual after all. If I refuse to allow my body to be used to support another life, that is my choice.
The scenario you've described is approximately the same as a suicide. Consider that people still do that even though it's illegal. Heinous shit will happen, but if you're genuinely interested in minimizing human death and suffering, safe and legal abortion is a necessary component to that.