Non-US carriers (except AC) do not belong to Pre-Check and do not process KTNs so you won't get Pre-Check on a BP from them. The only solution I know of is to reprint your BP at your US gateway, where you have to go land side anyway.

I don't know what actually happens but I would hope that if you have a KTN you will always get Pre-Check on a US airline BP but may get selected at security for a more thorough examination.