I am not even sure why we are talking about injuries here
The bottom line is that you didn't make it to the game until the second half. You were not there to work a complete game and shouldn't be paid for a complete game. You already know my opinion, you shouldn't accept pay for any of the game. It doesn't matter if the situation was out of your hands or not - you weren't there. And while I believe you when you say you were stuck in traffic, do you know how many times people use that excuse when they are late somewhere? Had I been in your situation I would not have accepted any money for the first game - even if I had only missed the tip off and got there before the first points were scored.
As someone else said - this doesn't make you a bad person or a bad official. Just accept it as part of life, learn from it and move on.