I have not had a child in residential care. Is there a social worker at the hospital you can talk to? If he's inpatient now most hospitals have a department for social workers help people get insurance or placement outside the hospital. It would be a good place to start.
In our state I applied for an SED waiver and social workers at the mental health facility help me apply. This is coverage through State health insurance, even if your child has coverage through his parents. The SED waiver (Severely Emotionally Disturbed) help to cover items that regular insurance or State insurance does not. At least that was the way it was about 10 years ago.
I hope you both can find help. It's hard doing this by yourself, especially when there's a parent that's not involved. Keep us updated. If you're not seeing the counselor for yourself you might try to get a referral. They can also help you navigate the system and be supportive of you.
Ksm