Batman clearly has PTSD. Why didn’t Alfred have him committed?

Alfred is old school butler. He don't see nuthin his masta do

No. Because it's strongly hinted (for those who pay attention...) that Alfred was secretly Mrs. Wayne's lover for years and that the both of them conspired to have Bruce's father killed, get their hands on his fortune, send the nipper away to boarding school or whatever, then drink champagne and merrily “do it” in every single room of the Wayne Manor for the rest of their lives (and you better believe the Wayne Manor is a huuuge estate: just ask Alfred, he's the poor sucka who's had to clean every room for the past 20 years... you never see any maids of other domestic workers in any of the films, do you?).

Anyways, unfortunately, the thug Alfred hired to do the job is a little overzealous and kills both Waynes. Now Alfred is stuck with the kid and has to take care of him until he turns eighteen. For a while, it's all cool for Al 'cause he's the legal tutor and has power of attorney over the trust fund, so he lives as high on the hog as he ever has (and Al' had done seen and try some crazy sh*t during his time in the service…) and brings hookers to "the Manor" every other night.
But when Bruce turns 18 and becomes the sole administrator of whatever remains of the Wayne fortune after 8 years of partying, Al's only chance to get his hands on the gold is if Bruce snuffs it. Al knows that. And Al being Al, he implants the crazy idea of dressing in tights and fighting crime in the kid’s head from an early age on (e.g. by leaving gory crime-scene pictures of his parents' murder lying around the place, laying bat-filled traps around the property for Bruce to fall into and develop PTSD, etc.).

Later on, Al also makes sure Bruce's "bat-costume" is as useless and stiff as possible (e.g. has nipples but can't turn the head, crotch area not reinforced so he has a good chance of becoming sterile from getting repeatedly kicked in the balls, thus nut leaving offspring to claim the fortune, etc.), and feeds him a bollocks code of ethics about not using lethal force, putting him at a clear disadvantage when fighting hardened criminals.

True story.

There was no reason to have him committed. Bruce wasn't a danger to anyone and in Alfred's eyes he channeled his energies into a constructive purpose. He was proud of the man Bruce had become.