It has been my experience that pitching is 90% mental after learning the mechanics. One must be able to focus during training, preparation and games. It appears what you are experiencing is an inability to focus on the task at hand because you are being sidetracked by the emotional attachment to the rivalry. This is completely natural, as any competitor would be "juiced" to face an old teammate or rival. The key however, is to channel that additional energy. I teach the kids I coach to take a deep breath (in and out) before every pitch to help clear their heads and settle their nerves. In addition, I suggest thinking of something amusing prior to stepping on the rubber. This may sound somewhat juvenile but it seems to assist them in focusing on the game situation and their most important job, which is executing the next pitch.
As for no fear or bulldog mentality, that may be something in you and it may not. The key is to be comfortable and confident. Find your mound personality and use it. I do not believe for one second that Joba doesn't get butterflies when he is facing Manny in a big spot. What I do believe is that he is confident in his preparation and focuses in on his execution.
In short, try to relax against your opponents regardless of who they are. You do not have to K the world to be successful. Prepare yourself mentally with a game plan and then execute it. Remember you cannot execute to your fullest potential when you are overexcited or too tense. Most importantly, enjoy your time on the mound! It is one of the few places in life where you control the tempo.