This earlier post of mine might partially address your question.

Despite trying to locate an answer in the primary materials, I've yet to see any solid evidence that suggests that it was a problem that Roman Christians needed to reconcile - they don't feel some sense of collective responsibility as "Romans" that needs to be dealt with or reconciled, because they attribute the crucifixion to Roman power embodied in the person of Pilate, and pre-Constantinian Christians don't identify with state-power anyway; there is also a strong and persistent trend to understand the agency of the Jewish leaders as responsible, and certainly that is at least half the gospel-portrayal of the event anyway. This does develop in some nasty anti-semitic directions.

tl, dr: Roman Christians didn't feel the need to reconcile this issue, and generally didn't write about it in this way.