If a mod uses AbstractClientPlayer.getLocationSkin() to, for example, render hands, it will use AbstractClientPlayer.locationSkin, which is the vanilla mc skin. A suggestion is to modify getLocationSkin() (and possibly getLocationCape()?) to use the texture that is currently in use for the model.

Examples:

I have the skin installed via the CPM model,

and if I take an item that uses getLocationSkin() to render hands, it will render the vanilla mc skin, instead of the CPM skin.

Or, in FTBUtilities, also uses getLocationSkin() for the head texture.

But, interestingly, the map in hand is rendered precisely with the model’s texture, although it also uses getLocationSkin(), there is probably a manual patch for this, but after looking through the cpm code I didn’t find it (maybe I was looking poorly)

Addition: сhanging getLocationCape() can fix the problem with the mod "Et Futurum Requiem" which ports elytras to 1.7.10, using getLocationCape() when render them

That'll just break the texture, in the gun example cpm has to replace both the texture and hand model, so patching these methods wouldn't work, and just break stuff. CPM models don't have to follow the same UV map as vanilla.
For the 'Et Futurum Requiem' issue, there will be an API to render custom parts in the future. Also cape and elytra textures are different in CPM.