I have been replaying the first scene to get a clearer impression of what happened and how it works.
In the scene you hear that one of her agents has turned traitor, murdered her top agent and knows the identities of many of her remaining agents so poses a threat to them. She wants the traitor killed quickly and efficiently to minimise the risk to her remaining agents.
Note -there is no option to question the traitor's guilt or call for a trial. The verdict of guilt is fixed - you can only influence the sentencing
your options
1) accept that you have no authority or accept you don't really know the details of the situation or accept that Leiliana probably knows what she's doing and do nothing.
2) Declare that all traitors must die
3) Object and call for the traitor's life to be spared. Leiliana responds by objecting to you challenging her authority and ability which gives further options
3a) tell her she's a rubbish spymaster 3b) say she's mistaken in this case or 3c) suggest there are more practical options than killing
Leiliana will object to the way you are putting the life of a single enemy ahead of the dozens of loyal agents who could end up dead as a result .
You get more options where you can say that the inquisition should be strict pacifists and never ever kill anyone (and then spend the rest of the game killing) or use your authority as leader of the inquisition despite not being the leader and having no authority.
If you pick option 1), 2) or 3c) Leiliana is irredeemably hardened and likely to kill anyone who opposes her regardless of whether they are threatening any lives or not.
3c) is on the bad path because to soften her you have to tell her that killing is morally wrong and that the lives of her agents are not important enough to justify killing an enemy to protect them.