I think he would have become a villain.
There was an interesting 'What If?' comic in the Marvel world which swapped Iron Man with Doctor Doom (mind swap), so Doctor Doom after having his harder past suddenly had the money and popularity of Iron Man... He goes power crazy and becomes a full on super villain, whilst Tony Stark, stuck in Doctor Doom's body has to work hard to climb back.
If Batman had been impoverished, and still had the trigger event that left him alone and stranded, with his natural inherited willpower he would have likely became a bat themed villain. Through the guidance of Alfred, and the mental training from Ras Al Ghul he channeled his negative experience and will power into good, whilst fighting the PTSD and mental scarring (with the bat obsession) that he got from the trauma. Without that guidance it's likely that the PTSD and the scarring would take over his life and personality, and his willwower and determination would push him to be a villain... seeking revenge against the human race for leaving him nothing... probably revenge against the rich and privalaged out of jealousy!