The oath generally includes supporting the Federal and State consitutions, the laws of the Nation, state and city, in that order. Now, as to the so called constitutionality of laws, cops do not have a personal opinion. By our legal tradition, laws enacted by the legislature and signed by the executive, are presumed to be constitutional until such time as a court of competent jurisdiction rules them unconstitutional. At that point, the cops will stop enforcing the law that is no longer valid.
For instance. A new state pre emption law went into effect in Ohio last year or so. It was enacted by the State legislature and applied state wide. The law basically forbade municipalities from enacting ordinances more restrictive than state law so as to promote uniformity throughout the state on firearms related matters. Prior to that time Cleveland had an ordinance that required local registration of handguns and forbade open carry. They interpreted printing under clothing as open carry unless you didn't have a license, then it was CCW. In short, the Mayor and law department sought to harass gang bangers and along the way refused to distinguish legitimate gun owners.
With the new law, the Cleveland Police publicly stated that they would not enforce the invalid ordinance even though ordered to continue to do so by the mayor. It did not matter whether any particular cop agreed or disagreed, only that the ordinance was in conflict with the higher state law and the cops refused to be sued for false arrest based upon the Mayor's whim.