Omega's number is chosen independently of the lottery number, however.

That's impossible (given your other specifications) against an agent who two-boxes iff the lottery number and Omega's number match. If the lottery number is prime, this causes Omega to ensure that a different prime number is placed in its box.

Edit: Realized; it should be "The lottery number is chosen independently of Omega's number."

Edit again: Just realized that this phrase sneaks in a causal postulate: Omega can't change the output of the lottery! Starting here in reasoning might make the problem a lot easier. In "logical time," first the lottery number is selected, then you make your decision, then Omega makes eirs. Of course, formalizing this is non-trivial.