Wish I had some good advice, it's very frustrating when they just seem "off." Your hen's face looks a little pale to me, but from the picture, her eyes look bright and alert.
I have an almost 4-year-old RIR who is also losing a little weight and doesn't seem to be eating well, but there's nothing obviously wrong with her externally. I have been resisting bringing her inside as I feel the same way you do- I will help them as much as I can when they're not well, but they need to live outside.
Is your hen a Golden Comet? My Golden Comet, who will be 3 this spring, went through a heavy molt this fall and her feathers came back in a much different color. Before she molted, she was truly golden colored, but after the molt, her feathers were much redder, sort of the color of your girl's head feathers. From the picture, your hen's feathers don't look unhealthy to me, but it is strange that they can change color so noticably.