I thought about why Stoneheart owuld keep Jaime alive and came to a theory, even if very unlikely: Stoneheart wants the death of all the participants of the RW, plus revenge against the Lannisters. If she kills Jaime, she will kill one of the plotters (that's what she thinks anyway, thanks to Roose Bolton). If she keep him alive, she could use him as a hostage to send Brienne on some errands and could gain some information about the different hosts along the Riverlands, since Jaime was the commander and knows how strong they are and what are the commanders' personalities. With that information, tha BwB could rescue the prisioners from the RW, take Darry, Riverrun, Harrenhall and maybe even rescue Edmure. So, if she is not just a vengeful spirit, but also has some cunning in her, she could use a live Kingslayer, instead of just feeding a few crows.