Okay say for what ever reason Bruce Wayne became incapacitated and his Batman family either moved on from being a super hero finding a normal life, moved on to another city vowing never to return to Gotham or died in the line of duty. Then years a later a new Batman appeared similar to Oliver Queen in the TV series Arrow like in the first few issues he's ruthless in his defending of justice in Gotham and even kills but only when all else fails (like when a mad man has an atomic bomb ready to go off in the center of Gotham and the shut off device is connected to his vital signs meaning the only way to shut off the bomb is to kill the bomber) and will torture a psychotic killer to save a life (like a child molester buries a child alive and the only way to save the child is to torture the truth out of the child molester).
And in his own narration this new Batman explains his actions with "I wont kill or torture needlessly but if it saves lives I will do whatever it takes to protect my city and those that live in it."
Then years later Bruce awakens to find that Gotham city hasn't changed much but at the same time he learns his replacement has saved it and protected as well as he did despite being slightly more violent then Bruce. What would Bruce do would he shut down this new Batman or take him under his wing and teach him to not be so extreme?
I would like to point out that this new Batman isn't a Punisher vigilante he would only kill and or torture when the situation was the most extreme in truth he's the type that tries to avoid those situations as much as possible and would later be found at a church asking God for forgiveness.