As a general principle "I'll kill myself unless X" is not a particularly good reason to do X. In fact it seems to be becoming a more prevalent type of threat as far as I can see. Now there are people with personality disorders (like BPD) who may be more prone to such sentiments, and if that's so then the solution to this needs to also be more nuanced than just "do what they want". Temporary insanity is probably a loose term, but in cases where a person is at risk to harm themselves and others, I'm not sure an abortion is the most obvious answer as you do. That's sort of like saying if someone poses a risk to someone else we should eliminate that someone else preemptively to remove the threat of harm, like in Speed (shoot the hostage).
You are not understanding the situation. You seem to keep thinking that the suicidal thoughts and tendencies are coming from some other source, such as personality disorders. But that is not the case.
In some cases (perhaps many), the cause of the suicidal thoughts is the pregnancy. There is no doubt about it. When you have a person who has had no such thoughts in the past, and suddenly becomes very serious about it when she becomes pregnant, there is good reason to believe the correlation is causation when it happens again and again and again.
That's just a truism: the bad thing caused in circumstance X occurs during circumstance X. It says nothing about causation, merely about timing. You can take a packed up box of dynamite, and when a gorilla sits on it you may have circumstantial reason to suggest that gorillas may be the cause of major explosions. But I trust you can see that this really tells us nothing about gorillas, but rather than there's a system in play that was not previously acknowledged.
If there was a drug that people were taking, and the leading cause of death of those taking it was suicide, wouldn't you consider it a strong possibility that the drug was causing at least some of those suicides?
Are you likening a natural biological process to taking experimental drugs? Granted the body is a drug factory of sorts, but this seems like an otherwise tenuous analogy.
This is not to say that all women who feel suicidal should automatically be allowed to have an abortion. But it does pretty well dispel the idea that the pregnancy has nothing to do with those suicidal feelings and ideas. So it cannot be dismissed as being from other causes.
You can quote me above where I said pregnancy had no relation to suicidal ideation. What I said was that a person threatening (or wanting) suicide is not ipso facto a reason to accede to a request. It is, however, a cause for real concern that should be taken seriously.
And if the pregnancy is causing serious intentions of suicide, then that affects the intrinsic right of the fetus to live. If the fetus' presence is causing a dangerous situation, the fetus' life may become secondary.
You are torturing the word "causing" here, to understate the point. There may be many factors, not the least of which are the mother's preconceptions about pregnancy, the society and its narratives, and the physical conditions the mother expects to encounter. None of these exists in a vacuum, and they are totally unrelated to the physical fact of the pregnancy itself. You would have to argue that the suicidal ideation is 100% biochemical (good luck with that argument, we won't have this kind of science for 500 years), and even if we granted this you still have no credible basis for the premise that someone's life becomes secondary
even if their presence causes danger for someone else. For instance take a famine situation, or some other zero-sum environment: would you argue that the presence of an
extra person in a famine, which necessarily constitutes a mortal danger for others who may starve if this person eats, therefore establishes that the value of this 'extra' person's life becomes 'secondary'? I'd also like to parenthetically note that the term 'secondary' may make the matter sound cold and dry but recall we're talking (according to the argument) about a person. Not that you need to pepper your clauses with terms like "unfortunately" and "with bitter regret", but it would be nice if at least you thought them before choosing brevity.
It's like a castaway in a boat that is found by a passenger ship. The castaway has typhus, which would infect most of the passengers and crew of the ship. Is the ship obligated to rescue the sick castaway, with the almost certain knowledge that his presence will kill many of the passengers and crew? I suspect that maritime law would say no, and certainly not call it murder.
You have your analogy backward. The proper framing in this example would be that there's a castaway in the boat, and their presence
makes someone else sick. Now this framing (and thus the analogy) would be mired by the fact that typically you make someone else sick
because you are sick, which is not in evidence in the fetus case. So let's tweek it to be that a castaway is in the boat, and someone else present is
fatally allergic to them. Do you think it would be reasonable in this case for the person
with the allergy to have the right to declare the castaway's life as of 'secondary' importance? We might well imagine this extreme scenario could degenerate into a "me or you" choice, but luckily IRL there are other options.
Yet if the presence of a fetus is causing the mother to become mentally sick, these laws deem it murder if the mother tries to get rid of the cause of her illness. 
Certainly it is not the fetus' fault that he/she is making the mother ill. But the fetus is the cause.
No, a single thing cannot
cause someone to become mentally sick. We don't have this kind of analysis available. You can cite the pregnancy as triggering it, perhaps, but that is not causation. The pregnancy could have catalyzed something else already in play, etc etc. I'm surprised I have to point this out to you.
I'm reminded a legal case a lawyer friend told me about, where the presence of a fly in someone's water allegedly caused him such mental distress that he lost total control of his life, couldn't work, and was in therapy. Naturally he was suing for damages. While one could perhaps believe the plaintiff really did have these symptoms and wasn't malingering, it is obviously farcical to posit as the primary theory that the fly caused this man's life to spiral out of control.
When a doctor determines it is the fetus that is causing severe mental illness to the point of suicide, then abortion should be an option. To deny it is to condemn mothers, and often their fetuses, to death. And that would be putting the life of the fetus above that of the mother.
I'm not really sure why you are intent on framing this as a fetus causing a (presumably) biochemical problem, and yet insistent that it would require a surgical solution. Why not a biochemical one, if it's just a question of hormones? Unless you're willing to admit into evidence that the mother's
ideas can be contributory, in which case your argument falls into a world of trouble.
