word or phrase for pursuing a losing argument in a certain manner

cavil - to oppose by inconsequential, frivolous, or sham objections

There's also quibble, which is very similar. But I think that more strongly implies avoiding the main issue by focusing on minor details, so it doesn't work very well if the subject has already effectively ceded the main argument, and is simply stringing out nitpicky details (often it's little more than a way of showing that the person is unhappy about losing the argument).

One expression not yet suggested is grasping at straws. From Christine Ammer, The Facts on File Dictionary of Clichés, second edition (2006):

grasp at straws, to To make a hopeless effort to save oneself. The term comes from the ancient image of a drowning man clutching at insubstantial reeds in an attempt to save himself, and it was often put as to catch or clutch at straws. It appeared in print as early as the seventeenth century and soon was regarded as a proverb. ...

If you're embroiled in an argument and you begin to see that your position is untenable, you may experience the intellectual equivalent of drowning; and at that point, it's hard to resist grasping at straws—even the flimsiest or most insubstantial of arguments—to stay afloat.