[Zeke]

You have more or less analyzed the heat problem you have. Heat transfer occurs in 3 forms: radiation — you filter it; conduction — you block it; and convection — you stop it.

None of the suggestions will do much for the problem except the one saying leave the door closed. Therefore, you need more indirect natural light sources and possibly a fan or 2.

You could add some more LowE glass panels to the doors. Say one or 2 more sections with glass.