The best textual evidence for Ethan to support his inference that Queen Elizabeth thinks Erik of Sweden is needlessly vain is:
D: But that we are not to give you an answer until we have seen your person is so far from the thing itself that we never even considered such a thing.
This sentence implies that Queen Elizabeth finds Erik's expectation of a response contingent upon seeing him to be unnecessary and perhaps overly self-important, which aligns with the characterization of him as needlessly vain.