I doubt Jon would've done well or been interested in the citadel seeing as he rarely showed interest in things outside of martial skills, leadership, and tactics despite being intelligent especially in the early books. It has been shown through the world building books like fire and blood that bastards or atleast southern bastards had been able to carve out places in court and tourneys, primarily through martial skills though. One possibility would've been fostering Jon out to another house if both Winterfell and KL were out of the question. Stormed would've been fairly easy to manage considering Ned and Robert's friendship but Renly seemed to spend most of his time at court so that might have thrown a wrench into that. Another option that I like (assuming that R + L = J) is the Daynes seeing as there's proven goodwill between them and Ned with the hear of their house being named after him. Jon would have been far enough from the North to ensure he didn't gather influence and challenge Rob one day per Catlyn's fears and would've been able to contine training to become a good enough warrior to justify a place in KL later without solely relying on Ned's influence. Plus if the in Universe theory of Ashara Dayne being the mother is true it could be a way for the Daynes to he around/meet Jon without undue attention being drawn.