Can I log copilot time on this?
Very unlikely. Look at:
http://www.easa.eu.int/certification...t-06022012.pdf
which lists the King Air as a single-pilot type. There is a way out - if an
operator has an Ops Manual declaring that it operates the King Air with 2 pilots, then it is possible to log P2 time. I'm not sure that a private owner is eligible to declare an Ops Manual, but it would be an expensive exercise. Of course, with a commercial operator, both pilots would need CPL or better. And you have another problem: you say you are
offered a job, so if you are to be paid you will need to be a CPL holder. If you are a PPL holder, I think there is no way you can log the time. But - it could be good experience, unpaid, if you are able to resolve the safety of the runway performance.