What I did...

I log the CFI time
PIC time (Not including the dual PIC or CFI)
Dual time...everything that is dual no matter if is Dual or PIC/Dual

I had like about only 2 hours of difference...then explain to the airline that you did it that way because you want them to have better your total time to match that the PIC time.

Which in any case you do like Wuzzo...add a note explaining them that you have more PIC time because you have a lot of Dual/PIC and is not listed in AirlineApps.