Leather armour for humanoid mobs has sections (stripe on helmets, kneepads and waist area on leggings, soles of boots) which remain the same brown colour regardless of what colour the rest of armour is dyed. I expected this to work the same way for the leather horse armour. However, given that there is only one respective image for the leather horse armour item and entity model, any dye applied to the armour affects the colour of the saddle as well.

This can be easily confirmed by applying a dye to horse armour and comparing the colour of the saddle on the dyed armour and undyed armour in an image editor.

Unless this is intended, the apparent solution is to match how the humanoid leather armour works and split the sections that aren't meant to be dyed from the sections that are into separate image files.