There are a lot of possible variables here, but I'm going to take a stab and guess.
1. The HR person is probably not used to hiring pilots and that was the issue with the not knowing on the Captain or SIC.
2. Technically, I do not think you are able to log the time since this is Part 91. In part 135 they would have to list you as a required crew member on their air certificate and therefore you would also have to pass a 135 SIC checkride. I would still log all of the time that you are in the airplane as SIC.
3. Since it is Part 91 I am guessing that is most likely a low number of hours that they fly per year. The low hours will hold you back from getting to the left seat, in addition to other things.
4. Insurance minimum hour requirements to be PIC in a King Air 250 are 1000 PIC usually and maybe 500 PIC at minimum with some insurers.
5. How many hours do you have and in what types? Multi time, Multi Turbine, etc. A lot of insurers will credit time in other types towards the King Air.
6. If you have previous Multi Turbine/Jet time then getting to the left seat should not take long. If you do not have the time, my guess is that it would take you 2-3 years of flying for the company and a lot of goodwill before they are willing to PIC type you in the King Air 250 ($10,000) so that you can fly with their current PIC until you build up enough King Air PIC time to fly as the sole PIC.
Cliffs: SIC on a King Air 250 is probably going to pigeonhole your aviation career if you are looking to climb the ladder. If its not about climbing the ladder, the location, pay, and people are good then go for it!