Since you are the Stannis expert, and since I've never seen a meta on it: why won't Stannis let Mance hold his son? It sounds like the kind of last request prisoners often make in our world, yet Stannis could also say, "Well, he hasn't earned it." I'm not condemning the One True King here, but what do you think Stannis' reasoning is in this matter?
Here’s the thing: I love Stannis, I get Stannis, but the One True King does not understand these, how you say, Hoo-man Emoshuns. Or rather, he does, but everything about his life (from Proudwing and the death of his parents forward) has taught him that exposing his feels is asking for trouble. As such, he buries them deep, and it takes a lot to bring them out. Davos consistently pulls it off, as does Jon. But while Stannis explicitly respects Mance, that same connection isn’t there. I wouldn’t condemn him for it either, but we king’s men must admit that this is the less-attractive flipside of the stoicism and bluntness we love about him.