My experience is that a public facility is better than a school gym.  In many parts of Korea the municipality will build a gym and charge a small admission fee (up to 3500 won in Seoul, as little as 1000 in smaller cities).  At a school gym, the club has to pay rent and to play there, you have to sign up for the whole experience and your membership fee goes toward renting the gym and paying for monthly soju and flesh-eating.  Having said that, every public gym has clubs operating within it, too, and they still demand membership fees for all but the rent.  Those who don't join a club can often find the clubs operating like cliques to exclude outsiders, but bona fide outsiders like many of us on fora like these can sometimes be enough of a novelty to bypass that treatment.

Having wasted that discussion, I now find that there isn't a huge list of clubs for Dong-gu in Busan and actually all 6, plus the two listed for Jung-gu are in schools.  There is apparently one in each of Gyeongnam Girls' High and Middle Schools and they both seem to play 5-8AM!  There is one in Busan Middle School.  The closest (Choryang Elementary School) and the farthest (Seongnam Elementary School) are the only ones that happen to play in the evenings.  There is also Gwang Il Elementary School, which is just off to the southwest of the Station and they have clubs that play both in the early morning and the evening.