Syncing music from new computer, but everything else from iPhone

I have a new MacBook Pro (purchased in December, so not the very newest one) and an iPhone 4S running, sadly, iOS5. I'll soon be getting an iPhone 6, but in the meantime, I need to properly back up the 4S, but there's a problem.

The iMac that contains the 4S backup has gone bad. I've managed to retrieve the iTunes library off it, but the latest mobile sync files won't copy over and are probably corrupt. So what I'd like to do is put my complete music library from the ailing iMac on the new MacBook, then sync the phone in such a way that everything FROM the phone except the music is backed up, then sync the music files as I see fit from the complete version.

Is that possible? What's the safest way to back it up without risking losing things off the phone?