As others have said, simply providing a stable place to be around another child and adult will do leaps and bounds. But if possible I would work on helping him get into activities he can do on his own when he is dragged to mom’s work and told to sit quietly in a corner or sent to the disinterested aunt—because that’s bound to happen again.
Maybe try discovering if he likes reading and what sort of material? Children’s paperbacks can be found inexpensively; library cards are free either for a physical library or digitally. If you find he’s into reading, an on-sale Kindle could be an option that you regularly load up with Libby ebooks.
How about a small sketchbook and a few pencils if he’s artistically minded? He doesn’t need a whole kit, just a few pieces for him to pass the time. If he’s into games, perhaps teach him solitaire or get him a travel chess set. Maybe sports are more his things, then fantasy football or some such.
I think teaching and helping him develop some low-cost, unobtrusive coping mechanisms could really save his mental health in the long-term. It’s a bit grim having to suggest all these solitary activities for a child, but the reality is that he will find himself in these isolating situations again. Giving him the tools to develop interests and hobbies without his mother/aunt’s intervention will put him in a better psychological place as he ages and will provide naturally-occurring encounters with potential friends who share his interests.
I was sent to a series of the ‘better’ programs—i.e., the ones without deaths attached to them. It took a decade of therapy to deal with the worst of the trauma and I still experience PTSD episodes several times each year. It’s astounding how these facilities carry on functioning with merely a name change. There’s almost no regulation, let alone patient rights. My first program was truly one of the better ones and they had patient rights information posted through the facility and hard copies provided in your welcome packet. When I was sent to my second program (wilderness), I asked what my patient rights were. I received dumbfounded stares followed by laughter.