= Solution
Let $\Upsilon_{123}=i\gamma_1\gamma_2\gamma_3$. Reversing three mutually anticommuting Majoranas changes sign, so
$$
\Upsilon_{123}^\dagger=\Upsilon_{123},
\qquad
\Upsilon_{123}^2=1.
$$
Moving $\gamma_4$ through the three factors gives
$$
\{\Upsilon_{123},\gamma_4\}=0.
$$
Thus $\Upsilon_{123}$ and $\gamma_4$ obey exactly the algebra of two <Majorana fermion operators>. Their exchange is implemented by
$$
\boxed{e^{\pi\Upsilon_{123}\gamma_4/4},}
$$
which conjugates one into the other, up to the orientation sign, just like a <Majorana braiding operator>.
Ordinary braids permute the elementary $\gamma_j$ with signs. Part (d) implements $e^{i\pi\gamma_1\gamma_2\gamma_3\gamma_4/4}$ by braids and parity measurements, and conjugation by this unitary maps an elementary Majorana to a Hermitian cubic monomial of the other three. Repeating this operation grows or shrinks an odd monomial by two factors, while ordinary braids place the desired indices in the active positions. By induction, every Hermitian odd monomial in $\gamma_1,\ldots,\gamma_{2M-2}$ can be reached from $\gamma_1$, with factors of $i$ inserted according to its degree to make it Hermitian. Hence braids plus suitable fermion-parity measurements map, in their action on $|\psi\rangle$,
$$
\boxed{\gamma_1\longmapsto
\text{any Hermitian fermion-parity-odd Majorana monomial}.}
$$
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