The issue is Batman hangs out with the JLA because he has plot armor. Specifically, he either has played 4D chess with everyone (ally and enemy), has the proper gadget, or both. Batman doesn't fight Superman like he's punching the Joker, he used kryptonite and anti-Superman armor, plus his array of vehicles, supercomputers, and weapons.
Batman isn't a fighter or a rogue, he's a high-level artificer. Or at best he's enough fiat to get whatever magical item he needs from the DM. Either way, he's not going to survive going toe to toe with anyone but other street-level heroes and villains. It's why in the classic game of "who would win" Batman always has to be qualified with how much knowledge and prep time he has.
That's not too say Batman doesn't contribute to the JLA, he's the world's greatest detective (when the plot allows it) with the world's greatest superpower (money). He can fight like a peek human, and has mastered several social skills. But outside of Gotham, he's sorely outclassed without his "magic" items and plot coupons.
Batman isn't all that hard to model, all you need is a DM willing to provide you with the ultimate backstory and magic item:
Backstory:
You are ludicrously wealthy to the point gold has little meaning. You can procure ANY item (magical or otherwise) as needed. Size is no object, you need a castle and plot of land? You get a castle and plot of land. Artifacts MAY take a bit longer, but you can procure them with time(or just use your utility belt, below).
You can also use your wealth for any other purpose required, such as (but not limited to) acquiring information as needed.
Magic Item:
Utility belt: Produce ANY magic item as a bonus action. All requirements to use the item are waived as to you.
I'm being a bit tongue in cheek - but not much!