Whilst it's all possible according to regs I hate to say that I think you insurance underwriter is going to be your big man on this one.
I know of a CJ1+ that is operated by a PPL holder. Having revalidated his class rating a number of times (and therefore being familiar with his insurance documents) I can confirm that his particular underwriter is happy to insure the aircraft with him as a named pilot.
I would also say that his handling skills and operational knowledge and technique are the equal (in fact better in some cases) of many professional licence holders who I have examined in CJs over the years.
So yes, it is possible, but make sure you get the right training!
The aircraft is easy to fly but the operating environment and performance require a
disciplined approach. Pilots who have some commercial experience will (hopefully) have had some discipline instilled into their operational approach by their employers. Some pilots are lucky enough to have this discipline occur naturally, as in the gentleman alluded to above.