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, and loops with built-in bounds reflecting $n > 2 \cdot 10^8,$ you will find the winner. NOTE: it turns out that this was true: since $$(2 \cdot 3 \cdot 5 \cdot 7 \cdot 11 \cdot 13 \cdot 17 \cdot 19)^2 \approx 9.4 \cdot 10^{13}$$ and we found out we could do slightly better than $10^{13},$ and any prime we use has an exponent at least $2,$ it follows that we do not need to use any prime bigger than $17.$
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 $p(1) = 1.$
NOTE: I see no hope of finding $m(6)$ unless we can prove that $p(m(6)) = m(5),$ and it is still a stretch in that case.
BEST
$$ 2^9 3^6 5^4 7^3 11^2 \approx e^{29.9013} \approx 9.7 \cdot 10^{12}$$
MUCH more careful computer run. No specific bounds on the exponents, just the requirement that the resulting product $n < e^{37},$ done using the logarithms. My original guess that $6$ was a reasonable upper bound on the exponents was just that, a guess, and not a good one. Mostly, you don't get enough benefit from extra large exponents on 2, and you don't get enough benefit out of $p(n)$ moving from 1296 to 2304 or 3600
log n p(n) 2 3 5 7 11 13 17 19
-------------------------------------------------------------
29.9013 1296 9 6 4 3 2 0 0 0
29.9966 1296 8 6 3 3 3 0 0 0
30.1019 1296 9 4 4 3 3 0 0 0
30.2197 1296 6 6 4 3 3 0 0 0
30.2703 1296 9 6 6 4 0 0 0 0
30.3472 1296 9 9 4 4 0 0 0 0
30.3713 1296 12 6 3 3 2 0 0 0
30.4038 1296 12 6 6 3 0 0 0 0
30.4808 1296 12 9 4 3 0 0 0 0
30.4891 1296 9 8 3 3 2 0 0 0
30.5216 1296 9 8 6 3 0 0 0 0
30.5719 1296 12 4 3 3 3 0 0 0
31.0407 1296 6 6 6 3 2 0 0 0
31.1742 1296 9 6 6 2 2 0 0 0
31.2245 1296 9 4 3 3 2 2 0 0
31.2512 1296 9 9 4 2 2 0 0 0
31.3423 1296 6 6 3 3 2 2 0 0
31.3438 1296 18 6 4 3 0 0 0 0
31.4758 1296 9 6 3 2 2 2 0 0
31.543 1296 6 4 3 3 3 2 0 0
31.6439 1296 16 9 3 3 0 0 0 0
31.7212 1296 12 9 3 2 2 0 0 0
31.7537 1296 12 9 6 2 0 0 0 0
31.8306 1296 8 3 3 3 3 2 0 0
31.9316 1296 18 8 3 3 0 0 0 0
32.0826 1296 6 6 6 6 0 0 0 0
32.1672 1296 12 12 3 3 0 0 0 0
32.2053 1296 12 3 3 3 2 2 0 0
32.2459 1296 16 3 3 3 3 0 0 0
32.3329 1296 18 4 3 3 2 0 0 0
32.4662 1296 9 6 4 3 2 1 0 0
32.5615 1296 8 6 3 3 3 1 0 0
32.5842 1296 18 6 3 2 2 0 0 0
32.6168 1296 18 6 6 2 0 0 0 0
32.6669 1296 9 4 4 3 3 1 0 0
32.6682 1296 9 6 6 4 1 0 0 0
32.6937 1296 18 9 4 2 0 0 0 0
32.7216 1296 4 4 3 3 3 3 0 0
32.7451 1296 9 9 4 4 1 0 0 0
32.7847 1296 6 6 4 3 3 1 0 0
32.8017 1296 12 6 6 3 1 0 0 0
32.8787 1296 12 9 4 3 1 0 0 0
32.8932 1296 9 9 8 2 0 0 0 0
32.9195 1296 9 8 6 3 1 0 0 0
32.9362 1296 12 6 3 3 2 1 0 0
33.054 1296 9 8 3 3 2 1 0 0
33.1369 1296 12 4 3 3 3 1 0 0
33.1622 1296 9 9 2 2 2 2 0 0
33.5519 2304 8 6 4 4 3 0 0 0
33.6057 1296 6 6 6 3 2 1 0 0
33.7129 1296 6 3 3 3 2 2 2 0
33.7392 1296 9 6 6 2 2 1 0 0
33.7417 1296 18 6 4 3 1 0 0 0
33.8032 2304 8 8 4 3 3 0 0 0
33.8161 1296 9 9 4 2 2 1 0 0
33.8464 1296 9 3 3 2 2 2 2 0
33.8933 1296 24 6 3 3 0 0 0 0
33.9266 2304 12 6 4 4 2 0 0 0
34.0418 1296 16 9 3 3 1 0 0 0
34.0444 2304 9 8 4 4 2 0 0 0
34.0577 1296 9 4 3 3 2 2 1 0
34.1273 2304 12 4 4 4 3 0 0 0
34.1516 1296 12 9 6 2 1 0 0 0
34.1755 1296 6 6 3 3 2 2 1 0
34.1779 2304 12 8 4 3 2 0 0 0
34.2706 1296 18 9 2 2 2 0 0 0
34.2862 1296 12 9 3 2 2 1 0 0
34.309 1296 9 6 3 2 2 2 1 0
34.3295 1296 18 8 3 3 1 0 0 0
34.338 2304 8 6 4 3 2 2 0 0
34.3729 2304 8 6 6 4 2 0 0 0
34.3762 1296 6 4 3 3 3 2 1 0
34.3801 1296 18 12 3 2 0 0 0 0
34.4183 1296 18 3 3 2 2 2 0 0
34.4457 2304 9 4 4 4 4 0 0 0
34.4805 1296 6 6 6 6 1 0 0 0
34.5387 2304 8 4 4 3 3 2 0 0
34.5469 2304 12 8 6 4 0 0 0 0
34.5635 2304 6 6 4 4 4 0 0 0
34.5651 1296 12 12 3 3 1 0 0 0
34.6242 2304 8 8 6 3 2 0 0 0
34.6639 1296 8 3 3 3 3 2 1 0
34.7245 1296 4 3 3 3 3 2 2 0
34.7533 2304 16 6 4 3 2 0 0 0
34.7799 2304 9 4 4 4 2 2 0 0
34.8109 1296 16 3 3 3 3 1 0 0
34.8976 2304 6 6 4 4 2 2 0 0
34.8979 1296 18 4 3 3 2 1 0 0
34.9134 2304 12 4 4 3 2 2 0 0
34.9258 2304 8 8 3 3 2 2 0 0
34.954 2304 16 4 4 3 3 0 0 0
35.0147 1296 18 6 6 2 1 0 0 0
35.0385 1296 12 3 3 3 2 2 1 0
35.0916 1296 18 9 4 2 1 0 0 0
35.0983 2304 6 4 4 4 3 2 0 0
35.1223 2304 16 6 6 4 0 0 0 0
35.1492 1296 18 6 3 2 2 1 0 0
35.1647 2304 12 6 4 2 2 2 0 0
35.1993 2304 16 9 4 4 0 0 0 0
35.2331 1296 9 6 6 4 1 1 0 0
35.2397 1296 18 9 8 0 0 0 0 0
35.2432 1296 24 9 3 2 0 0 0 0
35.2825 2304 9 8 4 2 2 2 0 0
35.2911 1296 9 9 8 2 1 0 0 0
35.2994 1296 9 6 4 3 2 1 1 0
35.3101 1296 9 9 4 4 1 1 0 0
35.3166 1296 18 12 6 0 0 0 0 0
35.3411 2304 16 8 3 3 2 0 0 0
35.3666 1296 12 6 6 3 1 1 0 0
35.3736 2304 16 8 6 3 0 0 0 0
35.3849 1296 27 4 4 3 0 0 0 0
35.3932 1296 24 3 3 3 2 0 0 0
35.3947 1296 8 6 3 3 3 1 1 0
35.4436 1296 12 9 4 3 1 1 0 0
35.4509 2304 12 8 6 2 2 0 0 0
35.4628 1296 16 9 9 0 0 0 0 0
35.4844 1296 9 8 6 3 1 1 0 0
35.4869 2304 18 8 4 4 0 0 0 0
35.5001 1296 9 4 4 3 3 1 1 0
35.5548 1296 4 4 3 3 3 3 1 0
35.611 2304 8 6 6 2 2 2 0 0
35.6179 1296 6 6 4 3 3 1 1 0
35.6362 1296 27 6 4 2 0 0 0 0
35.6662 2304 8 8 6 6 0 0 0 0
35.6864 2304 9 8 8 4 0 0 0 0
35.7225 2304 12 12 4 4 0 0 0 0
35.7525 2304 12 8 3 2 2 2 0 0
35.7694 1296 12 6 3 3 2 1 1 0
35.8199 2304 12 8 8 3 0 0 0 0
35.8872 1296 9 8 3 3 2 1 1 0
35.8883 2304 18 4 4 4 2 0 0 0
35.9701 1296 12 4 3 3 3 1 1 0
35.9861 1296 12 12 9 0 0 0 0 0
35.9954 1296 9 9 2 2 2 2 1 0
36.0263 2304 16 6 6 2 2 0 0 0
36.0765 2304 16 4 3 3 2 2 0 0
36.1032 2304 16 9 4 2 2 0 0 0
36.1169 2304 8 6 4 4 3 1 0 0
36.1797 1296 24 9 6 0 0 0 0 0
36.1978 2304 8 4 3 3 2 2 2 0
36.224 1296 27 8 3 2 0 0 0 0
36.277 2304 4 4 4 4 3 3 0 0
36.2911 1296 24 6 3 3 1 0 0 0
36.3066 1296 18 6 4 3 1 1 0 0
36.3279 2304 16 6 3 2 2 2 0 0
36.3682 2304 8 8 4 3 3 1 0 0
36.3909 2304 18 8 4 2 2 0 0 0
36.4209 2304 6 4 4 3 2 2 2 0
36.4389 1296 6 6 6 3 2 1 1 0
36.4491 2304 8 6 3 2 2 2 2 0
36.4916 2304 12 6 4 4 2 1 0 0
36.5477 3600 10 6 5 4 3 0 0 0
36.5492 2304 16 12 4 3 0 0 0 0
36.5545 2304 9 4 4 2 2 2 2 0
36.5724 1296 9 6 6 2 2 1 1 0
36.5903 2304 9 8 8 2 2 0 0 0
36.6068 1296 16 9 3 3 1 1 0 0
36.6094 2304 9 8 4 4 2 1 0 0
36.6253 1296 27 4 3 2 2 0 0 0
36.6265 2304 12 12 4 2 2 0 0 0
36.6494 1296 9 9 4 2 2 1 1 0
36.6573 1296 6 3 3 3 2 2 2 1
36.6722 2304 6 6 4 2 2 2 2 0
36.6922 2304 12 4 4 4 3 1 0 0
36.7166 1296 12 9 6 2 1 1 0 0
36.7429 2304 12 8 4 3 2 1 0 0
36.778 1296 18 12 3 2 1 0 0 0
36.7909 1296 9 3 3 2 2 2 2 1
36.799 3600 10 8 5 3 3 0 0 0
36.8353 3600 12 5 5 4 3 0 0 0
36.8356 1296 18 9 2 2 2 1 0 0
36.8944 1296 18 8 3 3 1 1 0 0
36.9379 2304 8 6 6 4 2 1 0 0
36.9448 2304 12 8 6 4 1 0 0 0