And this is the answer.
I just pulled Winston out of the back of the Detolf and looked at him under various lights. To the eye he's brown under indirect LED cool white light. But if you point the lamp directly at his head it picks out the red.
The camera also finds the red in him. Every time I've tried to photograph him I can't achieve a result that matches what I see myself. That's both with the flash on, and without the flash but with an LED lamp illuminating him.
The solution must be indirect, ambient light. Otherwise it's as though the camera sees through the surface to a reddish undercoat.
A similar issue became apparent photographing Ripley: her neck looked a completely different colour to her head when photographed. Yet in reality the difference is negligable.