= Solution
Replace the frame speed by $-2v$, requiring $|2v|<c$. With $\Gamma=(1-4\beta^2)^{-1/2}$,
$$
\boxed{\Delta t''_{AB}=\frac{\Gamma d}{c}(1+2\beta),\quad\Delta t''_{BC}=\frac{\Gamma d}{c}(1-2\beta),\quad\Delta t''_{AC}=\frac{2\Gamma d}{c}.}
$$
The outbound leg is now the longer one and the inbound leg the shorter one. Doubling the speed also changes the <Lorentz factor>; it is not enough to swap the two original leg times. A proposed doubled speed at or above $c$ does not define an inertial observer frame.
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