A plausible setup:

There are two equal-sized groups of large numbers:

• Prime numbers
• Composite numbers without any easy factorization

... such that you can't be expected to distinguish them in the allowed time.

The lottery works by picking at one of the numbers at random.

Omega's algorithm: for each number in those groups, predict whether you would two-box (given the already-determined lottery number). In the set of prime numbers for which you one-box, and composite numbers for which you two-box, pick a number at random and show it to you (so if it predicts you always two-box, it's sure to pick a composite number, etc.).

In that setup, if your algorithm is "if the number ends in 3, two-box, else one-box", then Omega will just pick a random number among the composites ending in 3 and the primes ending in 1, 3, 7, or 9.

And if your algorithm is "If the number is the same as the lottery, two-box, else, one-box", then yeah the chosen number will not be completely independent of the lottery number (Omega can only pick the lottery number if it's composite, or if it makes a prediction error), but that looks independent enough for the spirit of the post.