Short version is: it's (temporarily?) broken.
I'm an experienced Blender user and I have also tried many many ways to get textured or cycles materials into Blender.
As of now (Feb. 13, 2018) it doesn't appear that there are ANY methods to get "mesh color" from Blender 2.79 into Godot v3.
Based on a ton of testing, "Better Collada" is still the best working method at this point, but nothing, not Collada, not Better Collada, not gLTF by Khronos or Kupoman will take any form of texture or color (via Blender textures/png or Cycles materials via principled shader etc.) over into Godot.
So far as I can tell, as of today, this feature is totally broken - but Godot 3 has only been out for about 2 weeks so perhaps a fix is on the way?
I hope and pray the devs are aware of this fact - because everyone NEEDS meshes AND textures and/or Cycles' principled shader (and animations/actions) from Blender to easily transfer into Godot.
I did see a YouTube video here where a guy using Khronos gLTF ,2 shows an insanely complex import method, but none of us can reproduce his results; maybe it will work for you? Even if it did work, the process is such a train wreck that I can't fathom using it as part of the regular transfer process. Also I'm not sure if gLTF does animations (correctly), while I know Better Collada still supports animations in Godot 3.