[BODY_DETAIL_PLAN:BODY_HAIR_TISSUE_LAYERS:FEATHER]
If you go into b_detail_plan_default.txt and find the BODY_HAIR_TISSUE_LAYERS entry, it has a bunch of lines like this:
[BP_LAYERS:BY_CATEGORY:LEG:ARG1:1]
[BP_LAYERS:BY_CATEGORY:LEG_UPPER:ARG1:1]
[BP_LAYERS:BY_CATEGORY:LEG_LOWER:ARG1:1]
[BP_LAYERS:BY_CATEGORY:LEG_FRONT:ARG1:1]
[BP_LAYERS:BY_CATEGORY:LEG_REAR:ARG1:1]
Copy this entry to a new one with a different name (say, BODY_HAIR_LAYERS_NO_TAIL), and remove just the categories you don't want covered (deactivate them by removing the opening tag; see below). There's no need to remove ones the cockatrice doesn't have. Then go back into the creature raws and replace the old body hair plan with [BODY_DETAIL_PLAN:BODY_HAIR_LAYERS_NO_TAIL:FEATHER].
That will leave some parts covered in feathers, and some parts bare skin. To add scales on top of that bare skin requires some more work, and there are a couple of different ways of doing it. The easiest way is to paste another tissue over the skin by using the TISSUE_LAYER_OVER token. It's like making and attaching fingernails, but with slightly different syntax. The more complicated way is to create a new tissue for the scales (the existing one has the same properties as skin, which your creature already has), and use that as an argument in the BODY_HAIR_LAYERS_NO_TAIL plan. Say...
[TISSUE_NAME:scales:NP]
[TISSUE_MATERIAL:LOCAL_CREATURE_MAT:SCALE]
[SCARS]
[RELATIVE_THICKNESS:1]< 2 for more of an armour layer
[HEALING_RATE:100]
[CONNECTS]
[TISSUE_SHAPE:SCALES]
Then you'd go back to the custom body detail plan and change everything you want covered in scale to ARG2, and the line in the creature entry would change accordingly to [BODY_DETAIL_PLAN:BODY_HAIR_LAYERS_NO_TAIL:FEATHER:SCALE].
Additional question: If I define two different tissue layers is butchering going to return two types of cockatrice leather?
To do that you'd need two different material templates for the skin(s), possibly with two different material reaction products. If you put skin and scale on the same creature you'll get both as long as the creature's big enough, but scale is a separate material from skin and by default you can't make leather from it. If you mod it to be useable, then you'll get your use from it as if the skin doesn't exist.
[BODY:HUMANOID_ARMLESS:TAIL:2WINGS:2EYES:2LUNGS:HEART:GUTS:ORGANS:HUMANOID_JOINTS:THROAT:NECK:SPINE:BRAIN:SKULL:4TOES:BEAK:RIBCAGE:2HEAD_HORN]
[BODY_DETAIL_PLAN:STANDARD_MATERIALS]
[REMOVE_MATERIAL:HAIR]
[USE_MATERIAL_TEMPLATE:HORN:HORN_TEMPLATE]
[USE_MATERIAL_TEMPLATE:FEATHER:FEATHER_TEMPLATE]
[BODY_DETAIL_PLAN:STANDARD_TISSUES]
[REMOVE_TISSUE:HAIR]
[USE_TISSUE_TEMPLATE:HORN:HORN_TEMPLATE]
[USE_TISSUE_TEMPLATE:FEATHER:FEATHER_TEMPLATE]
[BODY_DETAIL_PLAN:TAIL_TISSUES]
[REMOVE_TISSUE:HAIR]
[REMOVE_TISSUE:FEATHER]
[USE_TISSUE_TEMPLATE:SCALE:SCALE_TEMPLATE]
[BODY_DETAIL_PLAN:VERTEBRATE_TISSUE_LAYERS:SKIN:FAT:MUSCLE:BONE:CARTILAGE]
[BODY_DETAIL_PLAN:BODY_HAIR_TISSUE_LAYERS:FEATHER]
What's in the TAIL_TISSUES body plan?
As it is now, the scale material isn't being used. It's being called ready for use, but it never gets added to the creature.