Welcome to FT!

You can't add the Korean Airways flight to your PNR. This leaves you at the mercy of the second airline in each direction should your first flight be delayed. For the sake of clarity:
(a) You can't have a PNR that contains an Avios flight and a flight purchased in cash (even if they are both BA or Oneworld, let alone outside of the happy camp of airlines!)
(b) You can't have a PNR that contains an Avios flight and another Avios flight if they weren't originally booked together, again irrespective of carrier.

I can understand the first but not the second. I don't make the rules!

My understanding is that that BA WILL NOT tag your bags beyond ICN but I'll leave those who know more to confirm.