An equivalent of $\neg ((a \wedge b) \Leftrightarrow c)$ based on a specific requirement
Outgoing from requirements can you  use $\oplus$? if yes, then, as we know, that $\neg(a\Leftrightarrow b) = a \oplus b$, then we can obtain
$$\neg ((a \wedge b) \Leftrightarrow c) = \neg (\neg a \lor\neg b) \oplus c$$
Update.
Accordingly to your update I wrote my one: not shorter, slightly different, but logically equivalent yours one
$$\neg \big[(a \wedge b) \Leftrightarrow c\big] = \\
=\neg \big[\big((a \wedge b) \Rightarrow c \big) \land \big(c \Rightarrow (a \wedge b)\big) \big]=\\
=\neg \big[ \big(\neg (a \wedge b) \lor c \big) \land \big( \neg c \lor (a \wedge b) \big) \big]=\\
=\neg \big[ \big(\neg a \lor\neg b \lor c  \big) \land \big(\neg c \lor \neg(\neg a \lor\neg b) \big) \big]=\\
= \neg\big(\neg a \lor\neg b \lor c \big) \lor \neg\big( \neg c \lor \neg(\neg a \lor\neg b) \big) $$