I know this thread's been dead for a while, but I just read through it and a couple things came to mind that I don't think were brought up:

Weren't the Apollo spacecraft constantly rotating during their translunar injection specifically so that no two sides were always in direct sunlight or shadow? It seems that alone would keep sunlight from keeping Apollo 13 warm during it's power outage.

Why, if Apollo was faked, wouldn't NASA just include more active thermal control systems if they didn't have to worry about actually building and implementing them? Why resort to passive systems if a.) they as you claim are inadequate and would uncover their fakery under close analysis and b.) they didn't have to actually plan for the weight and cost issues that so often influenced their decisions?

This is not an incident where one could say "it was impossible with the technology at the time" because more advanced active cooling systems existed and they chose the more passive system deliberately. Again, if it's so obvious a passive system is inadequate, why not just "use" an active system since they didn't have the burden of implementing it?