Heat pipes only reduce the thermal resistance between 2 points. They cannot cool something below ambient temperature.

Thermoelectric coolers do not compete with heat pipes. They are useful only when you want to obtain a temperature lower than the ambient temperature. Otherwise, heat pipes or liquid flow cooling are the right solutions.

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 was suggesting combining Peltier element _and_ heat pipes.