None of these are really what you're looking for, but it's the best I could come up with.
Yamato Nadeshiko Shichi Henge - I don’t think FL is actually mentally ill, but she’s definitely not “normal.” She’s severely antisocial, among other things. She definitely doesn’t go about life with a cheerful attitude, though. The drama is about her growth. ML isn’t rich, but I think SML might be? I don’t really remember their personalities that well. It has a very exaggerated style, but I thought it was charming.
Tatta Hitotsu no Koi - Female lead is in recovery from leukemia. I also think she has a pretty positive outlook on life. However, FL is rich, while ML is poor. ML also has a sick brother (cystic fibrosis I think). I’m not sure how I would describe ML’s personality. He feels like a real person, not a stock romantic lead.
My Holo Love - I think FL has prosopagnosia. But if I remember correctly, the plot is more focused on around the AI aspect than on her disorder. ML is either an AI or the programer who made the AI (I don’t remember that well).
Boku no Ita Jikan - ML is the one who is sick (ALS), but it definitely deals with themes like appreciating life through the hardships.