Depends on what you need the music for. Remember that when you use someone else's music in your fanfic you are using their poetry to convey a message. Using the right music is paramount.
Is it recommended listening whilst reading that passage? Is it for inspiration? To help set the tone? Is there a preferred length? I'm afraid I'm not familiar with your fiction but is there a preference for style? Length?
Avril's "When You're Gone" seems more a separation, pining and breakup type of song rather then one concerned with death, so I think we can strike that.
I just gave Remember Me by Josh Groban a listen and...
It's not exactly the kind of thing I'd consider poignant in a death scene or funeral type of setting.
While it remains your prerogative to use either of those songs you shortlisted, I'll throw in a short list of songs I think could work for either a Death scene or a Funeral / Remembrance setting.
Funeral / Remembrance
Two Steps from Hell - Love and Loss
Two Steps from Hell - Goodbye for Now
Two Steps from Hell - Tells a Story - Good for an upbeat ending where someone dies... strange but it works.
Two Steps from Hell - Freedom Fighters - Better as a remembrance then for a funerary type scene.
Fight and / or death Scene
Two Steps from Hell - Starvation - A good desperation feel to the song.
Two Steps from Hell - Blade Blade - A real glorious feel to this one... fitting towards a Dinobot type of character.
Two Steps from Hell - Heart of Courage
Two Steps from Hell - Forces of Destiny
Okay, I just realised I perhaps went a bit overboard on Two Steps from Hell... and on the post in general. I'll throw in a few mainstream songs and then leave it there because I could seriously dig for the next few hours that I, in all practicality, don't really have >_<
Sting - How Fragile We Are
Nightwish - Kuolema Tekee Taiteilijan
Nightwish - Dead Boy's Poem
Nightwish - The Islander
Queen - The Show Must Go On
Queen - Who Want's to Live Forever
And I'm forcing myself to stop right there. I could go on and on and on. Must. Stop. Grah!