Solution:
The answer is $3 + 5 + 7 = 15$. This is achievable since the three sides $\{3, 5, 7\}$ form the sides of a triangle.

To show it is best possible, note that $\{3, 5, 7\}$ are the three smallest odd primes; so any smaller triangle would need to have a side of length $2$. However, the two longer lengths would then be different odd primes; this would cause them to differ by at least $2$, which is impossible.