Well, I haven't read the book but this is my answer on why morality is necessary for happiness.
To be happy you need to have healthy relationships with others. This ranges from the simplest retail purchase all the way to marriage.
For buying a sandwich at Subway, it is required is that you, the cashier, and the owner agree on some basic property rights. Imagine trying to buy a sandwich from someone who believed they had a right to your money, but you deserved nothing in return. Would they make you a good sandwich? Would you ever give them money again? Similarly, imagine forcing someone to make a sandwich at gunpoint. Would they make you a good sandwich? Could you even eat it without fearing that they would spit in it or poison you?
Even for such a simple relationship to work, people have to deal as equals under a common framework. If it is impossible to even buy a sandwich without subscribing to a small portion of UPB, how much more morality is required to have a friendship or marriage that works. Not only do you have to agree not to use coercion, you also have to agree to honesty, openness, etc.