Clearly, their sum is greater than $2$, so the prime sum has to be odd. Since all three numbers can't be odd simultaneously, since their product is $320$, then two numbers are even and one is odd. There are exactly two odd divisors of $320 = 2^6 \cdot 5$: $1$ and $5$. Consider these cases.

$c = 1$, the following is possible:

$a = 2$, $b = 160$, $a+b+c = 163$ is prime.
$a = 4$, $b = 80$, $a+b+c = 85$ is not prime.
$a = 8$, $b = 40$, $a+b+c = 49$ is not prime.
$a = 16$, $b = 20$, $a+b+c = 37$ is prime and less than $163$.
$a = 32$, $b = 10$, $a+b+c = 43 > 37 \Rightarrow a+b+c > 37$.

$c = 5$, the following is possible:
$a = 2$, $b = 32$, $a+b+c = 39$ is not prime.
$a = 4$, $b = 16$, $a+b+c = 25$ is not prime.