If rationalists should "just win", and we equate winning with happiness,

Many of us don't, certainly not with happiness alone, but even if we did...

evidence shows that religious people are happier.

I accept a correlation between religious faith and happiness, but it's a long way from there to concluding that taking up religious faith is the best way to gain this happiness. Many sources of long-term happiness - sense of community, feelings of purpose, close family bonds, etc - are more likely to be seen in a religious person, but you don't have to turn to religion to experience them.