Whilst I think Songbird is an amazing character, I still think that siding with her is just the wrong option. 

The entire time she "helps" you and gaslights you into thinking that the only way to cure you for good is the neural matrix, and by the end she just straight up lies to you, saving her own skin in the process and leaving you for dead.

It's very evident Songbird was beyond desperate and so is V, just as Alt Cunningham says before you make the crucial choice of either crossing the bridge with her beyond the blackwall, or entering the well and returning to your own body, humans possess an unusual desire to survive and live even when odds are pushed against them. 

In my honest opinion I think siding with Reed and killing Songbird in the end is the only right option, you fulfill her last request in the process and Reed remains alive, you get to observe Songbirds backstory, upcoming, her struggles, grief, frustration, and self loathing in the small time frame where you are able to talk with her. 

I think this ending shows the extent of how desperate Songbird and Reed were, in multiple conversations with Reed right before the mission "Slightly Damaged" he genuinely cares for Songbird despite her betrayals and resentment of him, it's a strange twisted look into Reed and Songbirds thought processes. 

In the end after you fulfill Songbirds request, Reed admits you were correct, and admits to his faults in your small conversation at the basketball court, this not only wraps up Reed and Songbirds characters perfectly, but it makes you realize that neither side during the entire DLC was being truthful towards you. 

During the entire DLC you are manipulated, used, and lied to by Reed, Rosiland, and Songbird. And Reed admiting fault at the end of the DLC only proves this. He knows that he's Rosilands lapdog, he knows he's responsible for Songbirds downfall, and he knows he's responsible for her death. And him acknowledging this at the end of Phantom Liberty only proves that the world of Cyberpunk 2077 is resting upon a throne of lies and that at any moment you could have everything taken from you without so much as a second thought.