= Solution
Set $t_0=1$ and $t_i=-1$ for spatial indices. The <complex conjugation> of the scalar $i$ supplies a minus sign, while conjugating each of the three spatial <gamma matrices> supplies three more. In the original order,
$$
B^{-1}\gamma^{5*}B=-i(B^{-1}\gamma^{0*}B)(B^{-1}\gamma^{1*}B)(B^{-1}\gamma^{2*}B)(B^{-1}\gamma^{3*}B)=(-i)(-1)^3\gamma^0\gamma^1\gamma^2\gamma^3.
$$
Consequently the <chirality matrix> obeys
$$
\boxed{B^{-1}\gamma^{5*}B=\gamma^5.}
$$
No rearrangement of the <gamma matrices> is needed, so no additional <anticommutator> sign occurs. The star is entrywise <complex conjugation>, not <matrix transpose> or <Hermitian conjugation>.
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