A different, happier marriage might help him remain more stable.
I also think both Rudolph and Sissi might have been more stable if Rudolph's eldest sister hadn't died. So, when Sissi married FJ she was very young, inexperienced and not properly educated. Her mother-in-law, Archduchess Sophie, pretty much controlled her life in the first years of her marriage, basically took the children away from her because she was "too young to care for them", etc. Sissi was pushing back when she travelled with her husband to Hungary and took her daughters too against Sophie's advice. The kids got sick and the older one died. Sissi was devastated, felt guilty, stopped fighting Sophie for control of the children, her previous anxiety and melancholy got a lot worse and soon started her constant travelling. She didn't have another child for ten years after producing the male heir. FJ and Sophie wanted to toughen up Rudolph who was a sensitive child and got him a military officer for a tutor whose methods basically made him a nervous wreck by the time he was 9. At this point Sissi finally put his foot down and intervened, got him new tutors, etc. Then after a year or so, Maria Valeria was born and she got all of Sissi's attention, so much that she was suffocated with it.
Maybe without the whole trauma of the child's death and the guilt, Sissi, remained more stable, maybe had a couple more kids, didn't let the tutor abuse Rudolph for 3 years and Rudolph remained more stable due to not being traumatised with childhood abuse...