I've worked a lot of osage orange. From what I've seen, a summer in the sun will darken the wood about to the same point that stock is. I've never liked the results I got staining osage orange, though it can be done.
You'd probably have better luck matching the aged color of the wood you show by fuming the wood with ammonia. With this wood, I had best results at higher than usual temperatures (usually I want it less than 80 degrees F when fuming). I tented the wood and put the ammonia in shallow saucers beneath the objects being fumed. Then I let it sit 4 hours at ~95 degrees F before removing the tent. After airing out for about a week, I finished the wood as usual. In the years since, I've not heard any concerns about color change.