Complexity of Comparative Independent Set Decision Problem
This problem is indeed complete for the class of polynomial time Turing machines with access to an NP oracle with log many queries (also known as $\Theta_2^p$).  The result appears in a 2000 FST TCS paper by Spakowski and Vogel titled "$\Theta_2^p$-Completeness: A Classical Approach for New Results."  The proof presented there and a proof that I arrived at independently both rely on the $\Theta_2^p$-completeness criterion of Wagner ("Bounded Query Classes" SIAM Journal on Computing archive Volume 19 Issue 5, Oct. 1990).