So, if I were to run a restaurant, I would try a different strategy. I wouldn't label which products are vegetarian on the standard menu. Instead, I would also offer a vegetarian version that only included the vegetarian dishes for whomever might want it. The standard menu would of course also include the vegetarian dishes, just not singled out.
My hypothesis is that, as a result, non-vegetarians would become more likely to order dishes that happen to be vegetarian but aren't explicitly labelled as such, rather than filtering them out of their options by default.