Once you're using that commercial certificate, you're a professional. Don't accept the "valuable experience" crap as a justification for paying to fly the jet.
Not familiar with the Eclipse, but I'm pretty sure it's certified for Single pilot ops. Where does it say that 135 must be flown with an SIC for that jet? Is it an insurance requirement for your company or is it in your Ops spec? If it's not in the ops spec, you legally can't log that SIC time because the aircraft doesn't require two pilots. The same situation would apply to a caravan/baron operation.
Many places that operate single pilot aircraft like CJs find that it's much less expensive to slap a second pilot in the right seat than it is to insure and train a single pilot. The end result is a lot of guys flying around in the right seat logging time they really can't or being relegated to only logging the empty legs.