The decision for detention is not up to you. If the crime was committed against you and you are the victim, you have a say in the disposition of the case. Contact the victim advocate in your county, they are part of the State Attorney's office.
Willful and rebellious teens can be very difficult to raise, but locking your stepson up to "scare him straight" is not an option. The juvenile detention facilities operated by the government are for the purpose of housing children who have (1) committed serious acts of delinquency and (2) cannot (in the court's determination) be allowed a less restrictive environment. It is not a "scared straight" boot camp to scare your stepson into behaving. This is also not a "low-cost" option. Parents are legally required to reimburse the government when the child is housed in detention, in accordance with an "ability to pay" formula that is similar to child support. Given that you have a stepchild, you are probably aware that the government's definition of a parent's "ability to pay" is set by a mathematical formula, and most parents find it to be a very significant chunk of their income.
Even if the Juvenile Hall were to offer such a "scared straight" program, the consequences to the teen and your family could be far worse than you think. If you believe this teen is out of control, what do you imagine will happen if he spends a few months locked up with serious juvenile offenders like rapists, gang members, drug dealers and burglars? Wards in a juvenile facility are supervised, but determined teens can always find an unmonitored location to commit an assault or worse, and fights sometimes even happen direct view of staff. If your stepson makes enemies inside such a facility, he could be seriously injured or abused. Furthermore, after he has made friends with all of the local young gang members and drug dealers, he will be returning to your home.
You did not mention what this teen's unacceptable or dangerous decisions were. Since the teen is still placed in your home, I infer that either (1) the situation did not result in a report to law enforcement or (2) the local authorities did not deem the situation to require detention.
Don't assume that counseling isn't enough if you have not tried it. An experienced family therapist who has worked with families coping with rebellious or delinquent teens can work wonders. If your health insurance does not cover counseling, Seminole County offers services through its health department and through a variety of publicly funded or non-profit organizations.
Since the teen in question is a stepchild, you did not mention if co-parenting with the child's other parent is a realistic possibility. If the other parent continues to have a role in the young man's life, and also recognizes that his behavior is out of control, your family should discuss the possibility of working together to set up uniform expectations at both homes, and counseling between that parent and the child as well.
I practice law in the Eighteenth Judicial Circuit (Seminole and Brevard Counties) and have dealt with many juvenile cases here. As far as I am aware, there is no program that is truly similar to “Scared Straight.” The closest we have is “Reality Check.” The courts sometimes order it as a sanction in juvenile cases. You may wish to look into it. It is a far cry from Scared Straight but it’s as close as I’ve seen in the area.
I would generally advise against increased government involvement with juveniles, as you stand to irreparably harm them and taint their records. Moreover, you stand to permanently alienate the child and damage the relationship you have with your spouse. Matters that can reasonably be handled “at home” are generally better handled at home.
You cannot reasonably expect a request to have a youth placed into J.D.C. to be granted by the Court. They have no basis to honor it and they almost universally refuse to step in to raise one’s children when a parent experiences difficulty so doing.