How would you answer this question so that it can enlighten/delight/pacify someone from a non-Buddhist tradition
Notions of 'rebirth' are not considered related to enlightenment in the older scriptures. Also notions of 'rebirth' cannot lead to peace (pacification). The scriptures say notions similar to 'rebirth' have effluents (ignorant mental states), result in attachment/acquisition however can have the benefit of promoting merit/morality.
And what, bhikkhus, is right view that is affected by the taints, partaking of merit, ripening in the acquisitions? ‘There is what is given and what is offered and what is sacrificed; there is fruit and result of good and bad actions; there is this world and the other world; there is mother and father; there are beings who are reborn spontaneously; there are in the world good and virtuous recluses and brahmins who have realised for themselves by direct knowledge and declare this world and the other world.’ This is right view affected by taints, partaking of merit, ripening in the acquisitions.
The scriptures clearly unambiguously say it is the same person, the same man or woman or the same being that is 'reborn' according to their kamma.
Here, student, some woman or man is a killer of living beings, murderous, bloody-handed, given to blows and violence, merciless to living beings. Due to having performed and completed such kammas, on the dissolution of the body, after death, he reappears in a state of deprivation, in an unhappy destination, in perdition, in hell. If, on the dissolution of the body, after death, instead of his reappearing in a state of deprivation, in an unhappy destination, in perdition, in hell, he comes to the human state, he is short-lived wherever he is reborn. This is the way that leads to short life, that is to say, to be a killer of living beings, murderous, bloody-handed, given to blows and violence, merciless to living beings.
The famous Jataka Tales/Past Life Stories of the Buddha refer to the same person & people being reincarnated from life to life.
Suttas such as AN 3.15 report the Buddha saying about his past lives:
Now, monks, the thought may occur to you that the chariot maker on
that occasion was someone else, but it shouldn't be seen in that way.
I myself was the chariot maker on that occasion. I was skilled in
dealing with the crookedness, the faults, the flaws of wood. Now I am
a worthy one, rightly self-awakened, skilled in dealing with the
crookedness, faults, & flaws of bodily action; skilled in dealing with
the crookedness, faults, & flaws of verbal action; skilled in dealing
with the crookedness, faults & flaws of mental action.
Suttas such as MN 143 refer to the same (former householder earthling) person returning to earth from heaven as a god/angel to talk to the Buddha & The Venerable Sariputta.
"That is what the deva's son said. And [thinking], 'The Teacher has
approved of me,' he bowed down to me, circled me three times, and then
disappeared right there."
When this was said, Ven. Ananda said to the Blessed One, "Lord, that
must have been Anathapindika the deva's son. Anathapindika the
householder had supreme confidence in Ven. Sariputta."
"Very good, Ananda. Very good, to the extent that you have deduced
what can be arrived at through logic. That was Anathapindika the
deva's son, and no one else."
The doctrine of 'anatta' ('not-self') is not related to the doctrine of 'kamma'. When the Buddha taught the doctrine of good & bad kamma to unenlightened people the Buddha never mentioned anatta (not-self).
In summary, the Buddha always taught "a being" or "person" is "reborn" (example SN 42.3). If oneself was not assumed to be "reborn", an individual would have no incentive to do good & make merit.
In conclusion, we should not consider Buddhism to be special or always enlightened and different to the other superstitious religions. Buddhism is saturated with superstition. Therefore, when we try to convince rational people of the superstitions we believe in, we should consider it "suchness" they are disappointed, unsatisfied, incredulous & even fall on the floor rolling in laughter. We should have equanimity rather than have the craving expecting we can convince others.