First let's set the input sequence to be very malevolent toward Omega: it will always say the reverse of what Omega is projected to say based on the previous bits.

I'm not sure I buy this, but let me make sure I understand correctly. There is some super-super-intelligence generating bits, which is smarter than Omega and also antagonistic to it. That intelligence basically simulates Omega, figures out what Omega will guess, and sends the opposite value.

First, can't Omega just figure out that it's being simulated, and then find a source of pure randomness, and just guess according to that?

Second, can't Omega at least make sure it remains level with the pitiful human, by figuring out what the human will guess and copying that?

Third, how do you know there even exists a stream of bits that will make Omega lose constantly? It's not at all obvious to me that such strings exist. It seems like you're asserting the existence of N-bit strings that have Kolmogorov complexity substantially larger than N.