Wrong. There are two problems with this argument. First, even if the suffering of innocents did require us to believe in a prior existence, it would not require us to believe in reincarnation. The Mormons believe in a disembodied pre-existence, in which people were capable of sinning, but they do not believe in reincarnation.
Second, the suffering of innocents does not require us to believe in a pre-existence at all. Not all suffering is punishment for one's own sins (Jn 9:1-3). The prime example of this is Jesus, none of whose sufferings were for his own sins, for he had no sins (Heb 4:15).
Justice requires that everyone in the universe ultimately gets what he deserves. If a person has suffered unjustly he will be compensated for the pain endured. When we see innocent children suffering, it gives us just as much reason to postulate a future life in which they will be rewarded (i.e., in heaven) as it does a past life in which they sinned. The idea that God will compensate one who has been wronged is taught in Exodus 22:22-24, Malachi 3:5, and Matthew 5:11-12, 10:28-30.