I'm sorry, you're wrong. Bodybuilding tends to result in less functionality and than calisthenics and gymnastics. Gymnasts rarely ever move heavy weights, and look at them. And none of the bodybuilding exercises you mention are, in fact, necessary for a balanced physique.
Case in point, Eugene Sandow. His core training was the light dumbbell program he advocated in his books which he got from Professor Attila.
It might be worth his while for Bruce Wayne to do some bodybuilding training early on... when he's trying to build mass in a hurry. But at the end of the day, he's going to abandon that for more functional training that hardwires the qualities he needs for what he does. Gymnastics is likely to be a staple because of the nature of how he gets around the city, calisthenics, maybe kettlebells... but he's definitely not going to be putting the time it takes to look like a bodybuilder. There are many potentially dangerous weaknesses that tend to become ingrained in bodybuilders anyway.
And to be clear, we are defining bodybuilder here not with a handful of exercises that you point out above and taking into consideration football players or rugby players(who surely use some of the staples of bodybuilder routines in their training but use WAAAY more sport specific functional training). When we say bodybuilder we are talking about bodybuilders, not athletes. athletes are not bodybuilders and don't fully train like bodybuilders and to use them as an example for why you think a bodybuilder can do what Batman does is, frankly, dishonest. Football players and rugby players are not bodybuilders.