Yes, it does. It's not in one neat document like the US Constitution, but the UK does have a constitution and is a constitutional Monarchy.
The Oath of Coronation says nothing about ceding powers to intervene on moral or religious grounds either.
Yes, it does, and you've actually quoted it. You've focused on the first part of that section of the Oath - read the last part. I've put it in bold:
Will you to the utmost of your power maintain the Laws of God and the true profession of the Gospel? Will you to the utmost of your power maintain in the United Kingdom the Protestant Reformed Religion established by law? Will you maintain and preserve inviolably the settlement of the Church of England, and the doctrine, worship, discipline, and government thereof, as by law established in England? And will you preserve unto the Bishops and Clergy of England, and to the Churches there committed to their charge, all such rights and privileges, as by law do or shall appertain to them or any of them?
One of the rights and privileges of the Bishops referred to in the Oath is to sit in the House of Lords as the Lords Spiritual, and to vote in the House of Lords on matters of law.
The Queen has sworn to preserve that right. On that basis, she can't overrule a law that passed through Parliament, because the Lords Spiritual have already exercised any power the Church of England has to prevent a law being passed.
I spent much of my adult life in uniform serving Queen and country and am well versed in precisely how the constitutional monarch works in both the United Kingdom and Australia. I also used to hold a position of some authority (though not Holy Orders) in the Church of England, before deciding that Church had become too liberal for me - unlike the Queen, I have every right to leave that Church and go elsewhere. I actually do know what I am talking about when I talk about the Oaths, and how things work.
You are not saying that said rights preserved to the C of E’s bishops and clergy (nor is the oath) are ceded in perpetuity by the monarch. Nor is there an implication thereof. Royal prerogative still exists; only in the case of abolition of same would the monarch be duty-bound to rubber stamp laws that lead to the detriment of the kingdom.
And AFAICS, the Queen failed in the preservation of the first part of the oath I quoted.