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Your formula for SR is given in leading order of the ##v/c## expansion. You have to apply the same approximation also for the aether-theory result
$$\Delta t=\frac{2L v}{c^2} \frac{1}{1-v^2/(n^2 c^2)}=\frac{2L v}{c^2} (1+\mathcal{O}(v^2/c^2)),$$
i.e., you get the same leading-order result in SR an Aether theory.
$$\Delta t=\frac{2L v}{c^2} \frac{1}{1-v^2/(n^2 c^2)}=\frac{2L v}{c^2} (1+\mathcal{O}(v^2/c^2)),$$
i.e., you get the same leading-order result in SR an Aether theory.