Basil, I can think of lots of reasons in theory why a woman might want to have an abortion at 35 weeks. And like yours, they're often created by the sexist reality of life. But we then have to look at what would happen in theory in those cases if abortion on demand was legal.
People who are going through any kind of major medical event involving HCPs do not have complete bodily autonomy. The HCPs are involved in making decisions in (supposedly) the best interests of the patient.
In the case of the woman 30 weeks pregnant about to travel to an arranged marriage, a real case (who we don't know much about), what would happen if she turned up at 30 weeks because she had gone into early labour, or even at 40 weeks and went into labour and then had a still born baby? If she then expressed to HCPs that she was leaving hospital immediately so that she could go home, hide the fact she had ever been pregnant and get on a plane to another country where she also would not tell anyone, potentially to a rural setting with no health care, where she may not be allowed to leave the home or village (as frequently happens), to marry someone who she barely knows and is going to pretend to be a virgin to, the HCP would not go, okay fine. There would be a team of social workers involved, women from charities for British Asian women or a government department dealing with marriages like this, to try and get her to stay in the UK, move to a place of safety and continue receiving post natal care. It may be the case that if she refused post natal care that they felt she required, she could be sectioned and forced to receive it.
So if that woman turns up at 30 weeks and wants an abortion on the basis of this whole keep it a secret and fly to another continent plan, are HCPs going to, within any kind of ethical guidelines, agree to induce an abortion at that point, and be able to ethically decide that letting someone have a late abortion (when the woman has informed HCPs of their plan afterwards) that the woman is not at risk of death, disability and long term health problems if she undergoes such a major procedure under such circumstances.
The same would apply if she was from a religious family who didn't agree with transplants. Would a HCP agree to move a transplant forward by 10 weeks so that somebody could go home after major surgery and tell nobody about it then get on a plane. Surely there would be major ethical reasons to refuse the treatment at that point?
In your theoretical situation, the woman must have some conversation with the HCP about why she is having the abortion. If she gives the reasons you have said, what are HCPs going to do about it? They may do something vastly inappropriate (as you have outlined other government agencies are likely to do in response to the situation), she may get sectioned under some diagnosis because they think she is making stuff up or simply vindictive (according to them), she may be adequately treated by agencies actually dealing with the abusive man (unlikely, but possible). Whatever happens, there would be some process of counselling and consultation which would delay the process, also then putting her at risk of having the abusive man use the abortion request as reason for getting more control of the child if she does go on to have it. Whatever happens, it is not going to be as simple as the HCP just going, you want an abortion, go ahead, I'm not considering the ethics of what will physically or psychologically happen to you despite the fact that I have to do so in every other medical procedure.
Abortion on demand in principle would put late abortions on the same level as other medical procedures ethically. It would in theory give pregnant women the same level of autonomy as men having surgery. In reality it would probably put women seeking abortions at the same level of autonomy as pregnant women not seeking abortions but under NHS treatment for being pregnant. In theory it would involve reasonable ethical decision making by HCPS and in reality it would probably involve misogynistic and paternalistic ethical decision making by HCPS to the same extent that decisions about wanted pregnancies are made. Either way, abortion on demand in law does not actually mean the woman would be granted the abortion.
The theoretical case you mentioned is one where government agencies treat a woman inappropriately, and by legalising late abortion on demand, suddenly the NHS, another government agency, will agree to the abortion on the basis that it is going to behave appropriately despite operating in an inappropriate society and itself making inappropriate ethical decisions about birth all the time. I cannot see that she would be allowed an abortion, even if the legal framework allowed women to have one for their psychological well being, because the whole context of your example is that society does not recognise that her psychological wellbeing is at risk due to a DV situation. Another woman may get a late abortion under such a law, but I very much doubt this DV victim would.