The scenario you suggested is too complete. It is obvious that they should not have pressed on considering the poor weather you describe. If you wind the story back to the point where the weather was initially deteriorating then that is much more interesting as the decision is not so clear cut.
As an aside before the CRM issue, I believe that unplanned IMC should always be avoided where possible. Even with a fully capable aircraft and crew, in the conditions described, I would land and check my planning before going IF. With the crew you describe I think they should land and shut down.
Against what others have implied, in my opinion the cockpit is not a democracy. There is a captain and he/she has the overall responsibility for the safety of the aircraft. I don't agree with this ODV principle as different people have different opinions. Are you honestly saying that if you have 5000 hrs on type and have a new PPL holder in the LHS, who is nervous flying with 600ft cloudbase and 3km viz because he has never seen it before, and suggests that you should go home you would knock it on the head? I don't think so. As captain, you should take the concerns on board to formulate your own decisions.
IMO CRM is the flow of ideas/information across a cockpit to achieve a safe and successful sortie. In the indicated story there are two obvious issues for CRM. The first is the captain didn't listen hard enough but equally the LHS pilot did not express his concerns correctly, pretty much the standard problems with all CRM issues. Subtle questions like 'you do realise you're breaking the law here?' and 'Do you have an IR? (even if you know the answer)' and if it is getting really dangerous maybe 'if you continue to flout the rules I am going to report you (although not ideal for the CRM for the rest of the sortie, it MAY get the point across)'.
If I had to pick one person most at fault it would be the captain. Quite apart from not listening properly he should be dragging out as much information from the rest of his crew as possible to enable him(her) to make the best decisions.