I would have to specify a certain Batman before I could give an opinion on him. There are different versions. For the record I think the latest Batman movies suck rotten eggs. They go for gritty, dark story and turn Batman into basically a psycho who finds beneficial ways of expressing his psychoses. Kinda like Dexter - remember that show?
The original live action show Batman was of course cheesy, as were all of those shows at the time. The first Batman movie had a decent Batman. The Batman in the Justice League cartoon my brother was always watching was a bit too brooding and moody for my taste.
I think the best portrayal of Batman as he should be was in Batman: The Animated Series. He was more than a rich guy with gadgets fighting bad guys. He was doing a job that needed to be done. He saw being Batman as a responsibility, a service he was doing for the city he loved. Sure he used fear as a weapon, but it was always a means to an end, never an end to itself.
In every iteration of Hawkeye I have ever seen, he is just a guy shooting trick arrows. Can't even begin to compare to any Batman except maybe the very first cheesy one.