When I was in that boat, I just wrote a letter to the Chief Pilot at the regional, thanking her for her time and that of the recruitment staff. I also added some comments about how professional i thought the process was (and it was professional) and a few other bits of positive feedback. Nothing very long.

I was contacted by the CP thanking me for the feedback and large lead time. I later got hired by the major which owns that regional with no ill effects at all.