Democrats on Capitol Hill, together with labor groups and other allies, are waging an unusually aggressive campaign to derail the nomination of Andrew Puzder as labor secretary.

Mr. Puzder, chief executive of the company that franchises Hardee’s and Carl’s Jr. fast-food restaurants, has been a lightning rod as an outspoken foe of the Obama administration’s overtime regulation and the Affordable Care Act.

It is not clear how the Democrats intend to flip any Republican votes in the committee or in the full Senate, as they will need to do to defeat the nomination.

But if nothing else, the nomination battle has become a proxy for a larger struggle between labor and industry groups to shape public opinion on issues crucial to both.