I would say the timing has to do with Burke's career plans as an adult and John's desire to give him a " clean slate", and probable life insurance proceeds from Patsy's death. After all, he could have had very long standing policies on both of them which pre-dated her first bout with ovarian cancer.
How ironic would that be? Patsy's life insurance paying to get her off the hook for murder after her own death.
Someday, ALL of these people will face God and they will be judged for what they did to a beautiful tiny 6 year old!!!
Surely you don't think he really paid someone off with Patsy's life insurance money, do you???
And I guess you think he paid off the forensics lab in Virgina that did this DNA testing???