Find an approximate argmax using only approximate max queries
Extended comment of an idea or two toward a lower bound. Let $B = \Theta(\log n)$, say (though the best choice may be different), and let $\{v_1,\dots,v_n\} = \{\frac{1}{n}B, \dots, \frac{n-1}{n}B, B\}$. Consider drawing the input by picking a permutation of these values uniformly at random.

The idea should be that if we fix the indices of all values except for values $B$ and $\frac{n-1}{n}B$, then we should be able to show the difference in the algorithm's probability of picking one versus the other is very small: The variation distance between the results of the algorithms' queries is very small given the 50-50 distribution on assignments of these values to the two available indices and the results of any sequence of queries.

This argument holds for each pair of adjacent values, so we get a chain of constraints on the probability the algorithm picks the highest, second-highest, ... values. This gives an upper bound on the expected value of the algorithm, so we set that upper bound to $B-1$ and see what the number of queries has to be.

I couldn't improve on $\log n$ with the above approach yet, but I think you might get $(\log n)^2$ if you can leverage the fact that queries can't help with multiple steps at once. I.e. if a query changes when we move the highest value to a different index, then one of those times it it doesn't change when we move any other value to a different index.

Differential privacy might be helpful for one of these steps, e.g. if we only think about the case where we swap the location of the two highest values, the "sensitivity" of this query is just $\frac{B}{n}$ and then advanced composition might be helpful.

Sorry this is half-baked, but hope it can be useful!