A phrase for when one is reduced to quibbling on unimportant details having had to give up significant ground

Is there a popular phrase or idiom to capture the following very particular scenario? Person A and person B are arguing over something. Person A repeatedly gives ground to B until the argument becomes a matter of A trying to secure small details in favour of them, while having had to accept B's position.

The metaphor that I thought of for this was that someone threatens to kill you, but lets you choose at what time of day.

I realise that the phrase "winning the battle but losing the war" comes close to this, but I think there is a difference. There is the implication here that one's opponent is happy to allow one small victories having won the day. Obviously, a "hollow victory" is relevant here, but does not capture the particular cases I have in mind.

To give a common example:

When the UK political parties of the Conservatives and the Liberal Democrats formed a coalition, then the LDs were able to affect policy. But in doing so, they had to accept the fundamental outlook of the Conservatives. So they could haggle over the extent to which one or other benefit was cut, but in doing so they were conceding the very idea that there should be such cuts at all.

EDIT

I have found an idiom that comes pretty close: Rearranging the deckchairs on the Titanic.

This is not a perfect fit, since it means something more like "doing something that makes no difference because the major facts in the situation are already decided."

Perhaps the closest expression, rather than idiom, is something like "if you're arguing about detail x, then the other side has already won the argument".

1) trying for a consolation goal

(sports) A goal scored by the losing team near the end of a game, that has no effect on the outcome. — wiktionary.

Person A had long since won the match, and person B was reduced to trying for a late consolation goal [or score or prize].

2) Also a consolation prize has similar connotations.

Edit

3) The Lib Dems are fighting a losing battle.