I don't think that the Domestic Emoluments Clause's concept of "Emolument" goes so far as to apply to the death of a political opponent. Suppose that I'm a manager at a corporate job and the second-most likely candidate for a promotion. If I order a subordinate to kill the most likely candidate and I get promoted, I don't think that the prosecutor would think to add a "prohibited employment compensation" charge on top of the inevitable "conspiracy" and "first degree murder" charges.

(In a previous version of my comment, I thought that the previous commenter was referring to the Foreign Emoluments Clause and cluelessly asked whether there was a domestic version.)

Different emoluments clause: "he shall not receive within that Period any other Emolument from the United States, or any of them."

My bad. I have now edited my comment accordingly.

> If I order a subordinate to kill the most likely candidate and I get promoted, I don't think that the prosecutor would think to add a "prohibited employment compensation" charge

They would if you had used your company credit card to pay a hitman though. Perhaps this isn't the strongest argument for why the military for a political assassination isn't an official act. My point still stands though that all limitations of the powers of the President are still there, so not everything they do will be considered an official act.