OK, I am going to go against the other posters here. I am not looking to start an argument or anything, I just want to provide my input.
It seems as if your dad is trying quite hard. The fact that he is willing to go to therapy with you shows a lot. To me, it shows that he still loves you very much, and I think he will be totally fine with this once the shock wears off.
My recommendation is to keep the surgery scheduled but, tell your dad that you are willing to postpone it for a month if he absolutely needs you to. He did not ask for much, and since he has only known for about a week that is quite amazing. Work on getting him to go to some therapy with you, and when the time comes for the surgery if he really wants you to postpone for a month I suggest you do it. One month of waiting, to me, is well worth having your dad in your life forever.
I waited a month to start hormones for my parents. It helped them tremendously. Shortly after I started, to my surprise, they started to encourage me to go full time and change my name. They were extremely supportive. My mom told me that one thing that helped her was the fact that I took her feelings into consideration.
I'm not saying all parents are like this, but if you are lucky enough to have supportive family in your life you might want to do your best to help them through this. When you have people like that in your life your transition becomes about them almost as much as it is about you. Do your best to help them and you at the same time.
This is just my opinion, not fact, and it is in no way telling you how to live your life. Just some simple suggestions.