I dont see Batman doing Sneak Attacks. And Batman is kinda tanky. So I shy away from Rogue, despite the skills-fulness.
So far I am leaning toward Fighter Battlemaster. Some of his gadgetry can translate into Maneuvers and Weapon Masteries. He has the Unarmed Fighting Style, which for him is decent, light or no armor nor shield thus d8+Strength damage.
Batman is extremely Athletic, but within reallife olympic records. I would stat him in the master tier (levels 9 thru 12), at level 12, and Strength 18. Any level or ability higher than these, I consider superhuman and by definition magical. He has expertise in Athletics. Most of his ranged attacks are Thrown weapons, that actually use Strength for the Attack.
Relying on gish classes, including Ranger, Monk, and Artificer, requires strict reflavoring. A magical version isnt possible, because the archetype of Batman is that he is strictly nonmagical. He is a peak of humanity, not superhuman.
While his superpower is "rich", is humorous and kinda true, D&D Players Handbook only represents this by a Noble background, which isnt a great fit (History, Persuasion). Wealth by level has him wealthy at level 12, but doesnt convey the sense of trust fund. Batman is an example where the DM should step up to customize a background for the character concept. The general idea is he is a "Knight". His military training and experience emphasize Athletics. In my own games, I use the Nature skill for science, including physics, and by extension elemental alchemy. Thus Nature is for him to craft his own gadgets, and he has expertise in Nature as well. And a Smith toolset, to combo with Nature to craft the gadgets.