Wow, it seems as though you are contradicting yourself.. on the one hand you are saying that he is emotionally unhealthy and on the other hand you are saying that he is simply lazy so I am not sure what the real case is here if he truly has a mental illness or not. have you given up on him?
You say that he has many talents but has never attended any classes–is there a reason for that? perhaps he feels that it is too late to start anything?
perhaps, instead of just offering to send him to karate classes which may or may not interest him, you can do as I suggested and find several different brochures of programs that he may be interested in–maybe music since that is what he is into these days. again, dont offer to send him anywhere, just give him the brochures (of several different programs so he has options) and see if that sparks an interest in him. he may yet surprise you…
perhaps you can try some sort of reverse psychology by getting your other children to participate in extra-curricular non-threatening or non-religious activities like karate, kickboxing or swimming and when he sees how much they enjoy it he may be compelled to join them in their activities.