Van Cleef & Arpels, the century-old French jewelry firm, is the show’s primary sponsor and has supplied much of its megacarat menu: some 350 lavish pieces worn by royals, screen sirens and social swans. The extravagance isn’t the nauseous part; staggering displays of wealth don’t look out of place in this former Carnegie mansion. But allowing a luxury brand that’s still very much in existence to bankroll its own exhibition — one that often looks as if it were put together by the company’s creative directors — does not seem like a smart move, even if it draws A-listers to the opening-night gala. Some might argue that other museums have done the same thing less transparently: the Met with some of its Costume Institute shows or the Guggenheim with “Armani,” which was not officially sponsored by the designer but followed his pledge of $15 million. These events fool no one, and the Cooper-Hewitt, as a branch of the Smithsonian, should be especially leery. (The corporate connection will not, of course, protect “Set in Style” from going dark this weekend, along with the rest of the Smithsonian’s offerings, in the event of a government shutdown.) If “Set in Style” didn’t feel so in thrall to the company it might be less of an embarrassment for the museum. The show and its catalog, organized around rubrics like “innovation” and “transformation,” are full of breathless text that reads like ad copy. A short video supplied by Van Cleef & Arpels, showing workers cutting and polishing stones for the company’s signature Mystery Setting, engages in further brand-burnishing. Visitors, who have been lining up around the block, may not care much about any of this as long as they can enjoy some serious glitter. Alas, the show’s design makes looking a strenuous activity. The exhibition designers (Patrick Jouin and Sanjit Manku of the French studio Jouin Manku) have arranged the jewelry on long tables in the centers of the rooms — “a nod to the sumptuous dinners that the Carnegies held with world leaders,” a news release tells us, but the reality is less dignified. “It’s like a buffet,” said one viewer angling for a spot on a crowded Saturday. And the designers have added some distracting elements straight from the merchandising playbook: canopies of metallic leaves, handblown glass bubbles encasing the jewels. Meanwhile, the few sketches and other materials from the Van Cleef & Arpels archives are tucked away in corners, where no one is paying them much attention.