They want to solve the problem and get Joseph Kony arrested
Let me stop you right there.
Capturing Kony might have helped back when he was actually an important actor in the region. Currently, he is not even in Uganda. It is uncertain if he is even alive, and if he
is, he is an irrelevant figure with no more than a few hundred followers, and is likely somewhere in the DRC or Sudan. The Ugandan military pushed him and his followers out of the country in 2006. At this point, drawing attention to the issue serves two purposes: generating profits for IC, and boosting Kony's ego (if he is still alive); he is an irrelevance at this point, and treating him as a serious threat is possibly the worst thing to do, as it increases the chances of an attempt on his part to return to the spotlight, which would cause further suffering for the Acholi. Furthermore, what aid IC did send to Uganda at least in part went to supplying arms to the Ugandan military, who aren't exactly "good guys", either.
Quite apart from that: No, they don't want Kony caught. As long as he is still at large and at least possibly alive, he makes an effective bogeyman to center fundraising on.