“Bans don’t work,” he said. “They never have.”

Mr. Trump has also been under an intense lobbying campaign over the past seven weeks, waged by tobacco and vaping companies, along with conservative organizations, like Americans for Tax Reform, which are opposed to regulatory limits that would affect retailers, small businesses and adult consumers of e-cigarettes. Some have promoted enforcing sales restrictions to protect minors, or raising the national age to 21 for sales of all tobacco products.

The trajectory of the flavor ban — from a bold pronouncement of swift action to a fizzle after the political realities of taking such an action emerge — is similar to Mr. Trump’s stance on gun legislation. Months after back-to-back mass shootings in El Paso and in Dayton, Ohio, when Mr. Trump said he wanted to pass “very meaningful background checks,” warnings from gun rights advocates and Republican lawmakers about the political fallout that would result from doing that ultimately led to no action on the issue.

Inside the White House, the flavor ban has also become a proxy issue for how his advisers see Mr. Trump’s path to re-election. In one camp are those who believe he should try to win back suburban women, including mothers of teenagers who would presumably worry about their children becoming addicted to nicotine. In the other are those who advise him to assume that voting bloc would not favor him anyway and to focus only on energizing his base.

Mr. Parscale had flagged to Mr. Trump after he first announced his intention to ban most flavored e-cigarettes that it would hurt him with his base. Mr. Parscale and other advisers warned Mr. Trump to slow down, and announce he was going to take time studying the issue, telling him that a ban could depress turnout in critical states.

Those political concerns were not without merit: E-cigarette users have held protests outside the White House and outside Trump rallies that they may have attended under other circumstances. Protesters have also raised concerns about the potential closing of thousands of vape shops, which they said would hurt the economy and cost jobs across the country.

But it is not clear whether pro-vaping activists are one-issue voters.

While some advice to Mr. Trump was grounded in polling, some was based on a gut-level understanding of Trump voters: Taking away the right to smoke or vape would be something akin to taking away firearms.

In the opposing camp is Kellyanne Conway, a top White House adviser and Mr. Trump’s 2016 campaign manager, who has been telling colleagues and the president that it is a mistake to assume, as Mr. Parscale and others have done, that suburban moms who care deeply about a public health crisis for teenagers have deserted Mr. Trump for good.