Solution:

Suppose $2$ is one of the five primes. Then the total is even and bigger than $2,$ so it's composite. That means all five primes must be odd. The five smallest odd primes give $3 + 5 + 7 + 11 + 13 = 39 = 3 \cdot 13,$ which isn't prime. We can't hit $41$ with five distinct odd primes, but $3 + 5 + 7 + 11 + 17 = 43$ is prime. So the smallest such prime is $43,$ and its digit sum is $4 + 3 = 7.$ Thus, B is the correct answer.