Either way it will be good experience. You can log the time if you are a current and qualified Part 135 SIC, even if the aircraft is only single-pilot. Some say the time is garbage, and shouldn't be logged, but its really a matter of personal preference like logging Level D sim time. This requires and a checkride after completing the carrier's training program. You will be issued an 8410.
Do a search on 135 SIC and you'll find a LOT of discussion on the subject. If the operation is a IFR passenger-carrying operation under Part 135, then an SIC is required by FAR (135.101). Most (if not all) operators are given Operations Specifications paragraph A015 giving approval for Use of Autopilot in Leiu of a Second-in-Command (see also 135.105).
NoPax is correct. The FAA's position is that you can't even touch the radios if you are not a certified SIC and will be considered a passenger. Insurance has nothing to do with the regs. You can of course log PIC on legs where you are sole manipulator (repositioning, maintenance, etc.)