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.
Oh, if $h=0,$ for example, you do not multiply that in. The HC and superabundant numbers behave better in this regard. Put another way, your $p$ is almost multiplicative, but not unless you take care to build in $p(1) = 1.$
BEST
$$ 2^9 3^6 5^4 7^3 11^2 \approx e^{29.9013} \approx 9.7 \cdot 10^{12}$$
log n p(n) 2 3 5 7 11 13 17 19
-------------------------------------------------------------
33.054 1296 9 8 3 3 2 1 0 0
32.9362 1296 12 6 3 3 2 1 0 0
32.9195 1296 9 8 6 3 1 0 0 0
32.8932 1296 9 9 8 2 0 0 0 0
32.8787 1296 12 9 4 3 1 0 0 0
32.8017 1296 12 6 6 3 1 0 0 0
32.7847 1296 6 6 4 3 3 1 0 0
32.7451 1296 9 9 4 4 1 0 0 0
32.7216 1296 4 4 3 3 3 3 0 0
32.6937 1296 18 9 4 2 0 0 0 0
32.6682 1296 9 6 6 4 1 0 0 0
32.6669 1296 9 4 4 3 3 1 0 0
32.6168 1296 18 6 6 2 0 0 0 0
32.5842 1296 18 6 3 2 2 0 0 0
32.5615 1296 8 6 3 3 3 1 0 0
32.4662 1296 9 6 4 3 2 1 0 0
32.3329 1296 18 4 3 3 2 0 0 0
32.2459 1296 16 3 3 3 3 0 0 0
32.2053 1296 12 3 3 3 2 2 0 0
32.1672 1296 12 12 3 3 0 0 0 0
32.0826 1296 6 6 6 6 0 0 0 0
31.9316 1296 18 8 3 3 0 0 0 0
31.8306 1296 8 3 3 3 3 2 0 0
31.7537 1296 12 9 6 2 0 0 0 0
31.7212 1296 12 9 3 2 2 0 0 0
31.6439 1296 16 9 3 3 0 0 0 0
31.543 1296 6 4 3 3 3 2 0 0
31.4758 1296 9 6 3 2 2 2 0 0
31.3438 1296 18 6 4 3 0 0 0 0
31.3423 1296 6 6 3 3 2 2 0 0
31.2512 1296 9 9 4 2 2 0 0 0
31.2245 1296 9 4 3 3 2 2 0 0
31.1742 1296 9 6 6 2 2 0 0 0
31.0407 1296 6 6 6 3 2 0 0 0
30.5719 1296 12 4 3 3 3 0 0 0
30.5216 1296 9 8 6 3 0 0 0 0
30.4891 1296 9 8 3 3 2 0 0 0
30.4808 1296 12 9 4 3 0 0 0 0
30.4038 1296 12 6 6 3 0 0 0 0
30.3713 1296 12 6 3 3 2 0 0 0
30.3472 1296 9 9 4 4 0 0 0 0
30.2703 1296 9 6 6 4 0 0 0 0
30.2197 1296 6 6 4 3 3 0 0 0
30.1019 1296 9 4 4 3 3 0 0 0
29.9966 1296 8 6 3 3 3 0 0 0
29.9013 1296 9 6 4 3 2 0 0 0