Rob Pratt commented that for $n \le 10$ the optimal split without the requirement for equally sized parts is attainable with equally sized parts. This continues to hold up to $n = 28$$n = 30$:
1 1 [1]
2 2 [1, 4]
3 6 [2, 3, 4]
4 18 [2, 3, 4, 8]
5 30 [2, 3, 5, 7, 9]
6 576 [2, 4, 5, 6, 9, 10]
7 840 [3, 4, 5, 6, 7, 9, 13]
8 24480 [2, 4, 6, 8, 9, 10, 11, 12]
9 93696 [2, 4, 5, 7, 8, 10, 14, 15, 17]
10 800640 [3, 4, 5, 7, 8, 10, 13, 14, 15, 17]
11 7983360 [4, 5, 6, 7, 8, 9, 11, 12, 13, 17, 19]
12 65318400 [4, 5, 6, 7, 8, 9, 12, 14, 15, 16, 17, 19]
13 2286926400 [3, 5, 6, 7, 9, 11, 12, 13, 14, 15, 18, 21, 26]
14 13680979200 [3, 5, 6, 7, 9, 10, 11, 12, 15, 18, 20, 22, 23, 27]
15 797369149440 [3, 5, 6, 8, 9, 10, 12, 13, 15, 17, 18, 20, 25, 26, 27]
18 159943859712000 [1, 3, 5, 7, 8, 10, 11, 12, 13, 15, 16, 17, 20, 22, 24, 30, 31, 32, 33]
19 26453863460044800 [1, 3, 5, 6, 7, 8, 9, 12, 13, 14, 18, 19, 21, 24, 27, 28, 35, 36, 37, 38]
20 470500040794291200 [1, 3, 5, 6, 7, 9, 11, 12, 13, 15, 18, 19, 23, 24, 26, 27, 29, 31, 36, 37, 39]
21 20720967220237197312 [1, 3, 4, 7, 8, 9, 12, 13, 14, 16, 17, 18, 23, 24, 26, 27, 28, 29, 32, 36, 39, 41]
22 61690805562507264000 [1, 3, 5, 7, 10, 11, 13, 14, 15, 16, 17, 19, 20, 21, 22, 26, 28, 30, 31, 32, 34, 35, 42]
23 9203996481363478738944 [1, 3, 4, 6, 7, 9, 11, 12, 13, 17, 18, 19, 21, 26, 27, 29, 31, 33, 36, 37, 38, 41, 42, 43]
24 226577104515475594214400 [1, 3, 5, 6, 7, 9, 11, 12, 14, 17, 19, 21, 22, 23, 27, 28, 29, 33, 34, 35, 37, 38, 41, 42, 47]
25 4571875103611079835648000 [1, 5, 6, 7, 8, 10, 11, 12, 13, 14, 16, 19, 20, 22, 24, 26, 27, 29, 32, 36, 37, 39, 41, 43, 47, 48]
26 20218804109333464320000000 [1, 3, 5, 7, 9, 10, 13, 14, 15, 18, 19, 20, 21, 23, 24, 25, 26, 27, 29, 31, 36, 39, 41, 43, 46, 47, 52]
27 3678271958960426245017600000 [1, 3, 5, 7, 10, 11, 13, 14, 15, 17, 18, 19, 20, 22, 23, 25, 26, 29, 30, 31, 34, 39, 41, 43, 46, 48, 53, 54]
28 217018448461953024000491520000 [2, 4, 6, 8, 9, 11, 12, 15, 16, 18, 23, 24, 26, 27, 30, 32, 33, 36, 37, 38, 39, 43, 44, 45, 46, 48, 54, 55]
29 18646773190859199121234329600000 [1, 3, 5, 7, 10, 11, 13, 14, 15, 19, 20, 21, 22, 24, 25, 26, 28, 29, 30, 31, 35, 36, 38, 39, 42, 43, 48, 49, 52, 57]
30 90984319783113193178231808000000 [1, 3, 5, 7, 9, 10, 11, 14, 15, 18, 19, 20, 21, 22, 23, 27, 28, 31, 33, 35, 37, 41, 42, 43, 44, 46, 47, 49, 54, 55, 56]
Seva and David E Speyer address the question of asymptotics. The observation that for known values the difference neverrarely exceeds $n!$ ($n=29$ is the only exception, with a difference of $\sim 2.1 n!$) puts it on the level of Seva's abc-conjecture-conditional lower bound, and beats David's guess that the asymptotic is $\sqrt{(2n)!} e^{-o(n)}$. I have been no more successful than David in finding a strategy which beats $\sqrt{(2n)!}n^{-C}$, but I think there's an argument in the spirit of Erdős for why it's reasonable to hope that the asymptotic is $\Theta(n!)$ (or $\sqrt{(2n)!} e^{-cn}$).