I do not practice in Georgia but think it, like most states, has provisions for parents to seek voluntary services from the state child protection services (CPS) agency. "Giving a child up" though is not without consequences. For instance, they would be asked to comply with a service or treatment plan to help improve the child's chances of success in the future. If they didn't they could be found unfit or neglectful which might affect how they are treated if they came to the attention of CPS regarding their other children.
Rather than giving the child up, it would seem to me that a better goal would be to seek state assistance to get the child into treatment to address his learning, behavior and mental health issues. If the child won't go with the parents to be evaluated, they could seek court assistance through a child in need of services, family in need of services or person in need of services (whatever GA calls it -- CHINS, FINS, PINS) petition.
Discuss the issue with the child's pediatrician, school counselor or a family lawyer and figure out the best approach under the circumstances. I have linked the attorney find function of the National Association of Counsel for Children below.
I do not practice in Georgia, but I am sure that the state has an agency whose purpose is to protect children from neglect or abuse. Also, it is likely that your state has a court procedure where you can request help as a parent for a child who is willfully refusing your reasonable commands. This procedure and the protection procedure might enable your son to receive services to get him back on track. However, you may also lose control of the process to the state agency and/or the judge and things might not go the way you want them to go.
The best advice is to have your son evaluated by a psychologist or other mental health professional to see if emotional causes are the basis of his problems. If he is doing poorly in school, you can ask that he be evaluated by the public school to determine if he is eligible for special edcuation. Please note that a "Christian-based discipline school" may be counterproductive for a boy who migt have emotional problems or leraning disabilities or both.
I believe that parents armed with knowledge and understanding can do a better job raising kids than the government or the courts.