First, note that the setup is incompatible with Omega being a perfect predictor (you cannot possibly do the opposite of what the perfect predictor knows you will).

This is false. The setup is not incompatible with Omega being a perfect predictor. The fact that you cannot do the opposite of what the perfect predictor knows does not make the scenario with Omega incoherent because the scenario does not require that this has happened (or even could happen). Examining the scenario:

An evil Omega has locked you in a box. Inside, there is a bomb and a button. Omega informs you that in an hour the bomb will explode, unless you do the opposite of what Omega predicted you will do. Namely, press the button if it predicted you won't or vice versa. In that case, the bomb won't explode and the box will open, letting you free.

We have an assertion "X unless Y". Due to the information we have available about Y (the nature of Omega, etc) we can reason that Y is false. We then have "X unless false" which represents the same information as the assertion "X". Similar reasoning applies to anything of the form "IF false THEN Z". Z merely becomes irrelevant.

The scenario with Omega is not incoherent. It is merely trivial, inane and pointless. In fact, the first postcript ("PS. You have no chance to survive make your time.") more or less does all the (minimal) work of reasoning out the implications of the scenario for us.

Thus calling your sadistic jailor (SJ) Omega is misleading, so I won't.

I'm still wary of calling the Sadistic Jailor Omega even though the perfect prediction part works fine. Because Omega is supposed to be arbitrarily and limitedly benevolent, not pointlessly sadistic. When people make hypotheticals which require a superintelligence that is a dick they sometimes refer to "Omega's cousin X" or similar, a practice that appeals to me.