Here, have 2 headcanon's about the JSA and JL
In the earths where the JSA was the first hero team, then retired, then a while later came the JL as a new generation of heroes, the Hall Of Justice was actually JSA's original base of operations that was turned into a museum after the JSA officially retired/disbanded (even if you see some heroes still operating now and then) and they just 'gifted' the space to the JL after they were formed, sorta to continue tradition - though maybe their legacy heroes + their old members that joined the League may technically be the ones to own the building, with the JL being allowed to operate there bc said legacy heroes and former members are part of the League.
The JSA still have their original meeting room there though, and are known to visit often, to both gossip, mentor new heroes or even just to steal a mission or two to do it in the name of 'JSA reunion'
2. the JSA has beef with Batman or at least dislike him, this could be for whatever reason such as:
Based off of the previous headcanon: when they were being given the tour of the JL's version of the Hall Of Justice, they walked in on Batman directing a bunch of construction people around to change a lot of the building. He had to be talked down from it and the JSA were Offended because the HOJ was good as it was!
Dinah or even just Oliver has beef with Bruce, and they are siding with Dinah/Oliver because of relations, nothing personal Bruce
On the first meeting, Batman glared at Barry to shut him up, Jay took offence and declared a mostly one-sided aka only Jay's part beef with the Batfamily and the JSA sided with the Flash Family
Wally was told Bruce's secret identity Before Barry because he had befriended Dick, and Bruce made him keep it secret from other people. JSA took Offence
Bruce is in a romantic relationship with Diana or Barry or any of the legacy heroes and the JSA doesn't think he is good enough to be in said relationship
Bruce's contingency plans for them involves their family, and they Don't Like That Mr Dark Knight
Bruce said on live television he thought the name Justice Society was tacky, and possibly criticized them for having their secret identities public (if we go with them having them public, that is)
Bruce is convinced that they dislike him, they just think it's funny to give him the taste of how he makes others feel
