Well, make it an answer. The simple observation, also true for highly composite numbers (which resemble this problem) is that getting a large value of $p(n)$ means $n$ has non-increasing exponents in its prime factorization. Furthermore, no exponent is exactly $1.$
So, I am not entirely convinced that $p(m(5)) = 1296.$ Maybe, maybe not. However, i am convinced that if you go up to eight primes, $$ n = 2^a 3^b 5^c 7^d 11^e 13^f 17^g 19^h $$ with $a \geq b \geq c \geq d \geq e \geq f \geq g \geq h \geq 0$ and none exactly 1, say $a \leq 6,$ say, and loops with built-in bounds reflecting $n > 2 \cdot 10^8,$ you will find the winner.