You could have heat pipe filled with liquid that evaporates at 5 degrees. This way it would draw heat from ambient level temperature and lead it to peltier device that would cool it below 5 deg and liquefy it back again. This way you could have peltier in the middle of your thick insulation layer with heat pipes drawing the heat into it from the cooled space and drawing the heat from the other side of it outside (using traditional heat pipes this time).

I think you may have just reinvented absorption refrigeration, previously invented by Ferdinand Carré in 1858.

https://en.wikipedia.org/wiki/Absorption_refrigerator