Presently, you're logging empty (Part 91) legs only, in a turbojet, multi engine airplane, correct? You indicate that you have 60 hours in the aircraft, but can't log it (you have zero hours in the aircraft).
Do you have a type rating in this aircraft?
14 CFR 135.99(c)(3)(iii) requires that you comply with the crewmember testing requirements of Part 135, Subpart G, in order to participate in the pilot development program. That's not a logging requirement: that's a participation requirement.
135.293(b) requires you to pass a competency check in the aircraft. Without that, you can't participate in the pilot developoment program. You must undergo all the initial training and checking for that certificate holder, in order to participate in the pilot development program.
You seem to be caught in a Catch-22; you're not confident enough or comfortable enough to take a checkride, but you can't be in the pilot development program unless you do take that checkride.
I think it's fairly clear that you are not type rated in the aircraft. Type ratings, incidentaly, are conducted to ATP standards.
If you're not type rated in the airplane, you can't log PIC. If you're not in the PDP, you can't log SIC. It's a single pilot airplane, so your "dead" legs (part 91 legs) aren't loggable. You can't log it as sole manipulator PIC, because you're not qualified. You can't log it as SIC because there's nothing that would permit you to do so. It's single pilot, you're not under 135 on those legs, not qualified to participate in the PDP program, and unless the PIC is a flight instructor signing your time off as instruction received, then you have no provision that legally allows you to log that time.
The question is, then, what you are doing in that seat. If you don't plan to take a checkride as SIC in the pilot development program, do you plan to sit in that seat not logging time, indefinitely?
Think about the role of the SIC: it is to assist the PIC in normal operations, but also to take command of the aircraft if the PIC is incapacitated or otherwise unable to perform his or her job normally. That seems straightforward on a nice, calm, VFR day flight into a long runway. Are you ready to do it with ice and convective weather at night in the mountains to a stiff crosswind? If you're not, maybe you are in over your head.
Your commentary suggests that this is an hours-building exercise for you. It also suggests you have no way of building hours while doing this. If your goal is to gain experience, by which I mean both logged hours and actual life-experience on the job, then you may need to step back, get more qualifictions, and get a job that maches your experience level. Instructing is not a bad gig. It sounds like you have a fresh commercial and just enough hours to meet the minimums, so even a 135 VFR PIC position is out of the question.
You may want to look at instructing, flying jumpers, towing banners, or doing something along those lines, or finding a SIC program you can jump into presently that will operate at your speed and confidence level.