I ran into this too - it seems that the avionics will work for about the first five minutes - and then they just fade out even if there is external power.  I *think* this may have something to do with the "infinite battery life" option setting but don't quote me on that.

The only thing I can do is map a button or key to "Avionics on/off."  Once the engines are started and generators going, I flip the switch and everything works.  (An easy way to check this is to bring up the GPS - if it is "black" then you have no avionics).

No matter how hard I've tried, I cannot keep the avionics on on external power/battery alone for very long without losing them.  They won't stay on unless the generators are running.

This is probably something buried deep in the arcana of the FSX SDK - ya just have to work around it.

Brian