WMG / no-one has to die.

Christina will save her mother by giving Lionel the results he wanted from the TEMPEST experiment with her mother.

According to what Lionel told Christine in the first timeline where everyone survived, the TEMPEST machine cannot send a consciousness before the first time it was turned on, which was when her mother was sent back through time. Depending on how much time passed between the machine becoming active and her mother using it, she can bruteforce her way into that span of time, immediately contact Lionel or her mother, arrange a meeting, then give Lionel all the info she has from her dimension-hopping journey. Whether she can guarantee her family's safety after outing herself as the machine's "success story" is another matter entirely, but since she chose to kill the innocent guards and set Fenix Corp on fire in her original timeline (her crimes in alternate timelines notwithstanding), she might not be the only Karma Houdini.