Hi OP.
I'm autistic and work in a mental health field. I also still refer to myself as having Aspergers because that is my diagnosis, I'm comfortable with the term as it's better understood that just 'autism' so it's convenient for me to use it in certain situations. It's true that it's a term that is no longer used when diagnosing, for understandable reasons, but I don't think you should beat yourself up for using it either because it is still part of our lexicon.
As I said, I work in mental health myself, and I have professional dealings with people in your role, so I'm often in the same environments you work in and understand what you and your colleagues deal with as I work with the exact same people who are your patients. I know I could not do your job because my personality sounds very similar to your colleague's. All-night singing would have me in a raging temper, and I wouldn't react well to patients touching me either. For that reason I can empathise with your colleague, but that still doesn't mean that their actions are reactions are definitely down to being undiagnosed autistic. It certainly could be, because I think those aspects of my personality are inherently part of my autism, but it could equally be that your colleague is just a perfectly neurotypical person who isn't at all suited to their role.
I can understand your concern for them, and I think it's creditable that you clearly are thinking about their wellbeing and that of your patients, but I honestly do not think it's your role to go discussing autism or their actions with your colleague if you are neither their manager, their partner, their doctor, or a therapist that they themselves have approached. It's entirely possible your colleague has already been through the process of being assessed, and if they have not, they may well react negatively to your involvement, and it's not beyond the realms of possibility that a complaint could see you facing disciplinary action yourself.
Sometimes with people who genuinely are living with an undiagnosed condition it does take a third party making a well intentioned suggestion for a light to go on and lend some insight into what someone might be experiencing, but I don't believe that's an appropriate thing for you to be doing given that you are simply a colleague employed in the exact same role that they are. If anything, it might be appropriate to mention to whoever manages you both that you have some concerns about their behaviours, and if you are asked to elaborate you might want to mention that your suspicion that the negative behaviours perhaps have an explanation other than your colleague simply being uncaring or compassion fatigued.
Regardless, for the sake of your patients if nothing else, I think their behaviour does need addressed, but I don't think the way to begin going about that is for you to approach them directly about the possibility that they may be autistic.
Good luck whatever happens.