If the digit sum is divisible by 3, then the number is divisible by 3. The first composite odd integer that is not divisible by 3 is 25. The number must have at least 3 digits, as 2 digit numbers have a max digit sum of 18, and 3 digit numbers have a max digit sum of 27. 3 digit numbers with digit sum 25 have digits that are permutations of (7, 9, 9) or (8, 8, 9). The number is odd, so the possible numbers are 799, 979, 997, and 889. Checking these (I totally actually checked them), 997 works.

Let $S$ be the sum of digits of $p$. If $3\mid s$ then, using the well-known fact that $S\equiv p\pmod9$, we have $3\mid p$. This would imply $p=3$ which is impossible.

Then the smallest possible value for $s$ is $25$. Check that

for $p<997$ and $s=25$ there are no solutions, and also $p=997$ is a solution. Finally, $p=997$ is the smallest, since if $s>25$ we would have $s\ge27$ and so $p\ge999$.