In 霸王别姬 is 霸 just a name? I have not seen the film or read the book， I can't judge. 

Is this 霸 just a name, or does 霸王 really mean 'the tyrant king'？
It can be a general term, i.e. not a proper name, but it is/was often used to refer to a certain historic person called Xiang Yu. So 霸王 will not translate as "King Ba". 

Chinese Wikipedia:


  霸王是指比一般王还强盛的君王，时常是文人雅士对项羽的尊称。




More references:

Baidu Baike:  霸王（汉语词语）


  霸王，古时指霸和王，也用来尊称霸主，特指西楚霸王项羽，或者比喻非常蛮横的人。


In addition Baidu Baike lists 12 entries in total for this word.