If we do two one-sided tests instead of a two-sided test, can we get extra power by using a better multiple testing correction than Bonferroni? A standard problem in statistics would be something like the following.
$$
H_0: \theta = \theta_0
$$
$$
H_a: \theta \ne \theta_0
$$
This is a two-sided test.
What if we did the following two tests?
$$
_1H_0: \theta = \theta_0
$$
$$
_1H_a: \theta > \theta_0
$$
$$
_2H_0: \theta = \theta_0
$$
$$
_2H_a: \theta < \theta_0
$$
The complaint that I see is that this raises $\alpha$, since there are multiple testing issues at play, and then if we consider multiple tests, this becomes equivalent to the two-sided test.
There is more than one way to adjust a p-value, and that approach, to me, implies a simple Bonferroni correction.
There are better adjustments than Bonferroni.
Could we keep $\alpha$ at the level we want (say $0.05$ or $0.01$) but increase power by using a better correction than Bonferroni?

Yes. Assuming you have more than one such two-sided test, you may consider:
The Holm-Šidak adjustment gives much more powerful sequential rejection control over the family-wise error rate under dependency.
Holm, S. (1979). A simple sequentially rejective multiple test procedure. Scandinavian Journal of Statistics, 6(65–70), 1979.
The Benjamini-Hochberg adjustment gives even more powerful control of the false discovery rate.
Benjamini, Y., & Hochberg, Y. (1995). Controlling the False Discovery Rate: A Practical and Powerful Approach to Multiple Testing. Journal of the Royal Statistical Society. Series B (Methodological), 57(1), 289–300.
