Carlos Sada Solana, new Mexican ambassador to the United States, arrives to news conference in Mexico City, Mexico, April 21, 2016. REUTERS/Edgard Garrido

By Dave Graham

MEXICO CITY (Reuters) - At first, Mexico's government did its best to ignore Donald Trump. Then it likened him to Adolf Hitler. Now it has appointed a new ambassador to come up with a better plan.

Fed up with the front-runner for the Republican presidential nomination labeling Mexico as a cradle of drug-runners, job poachers and rapists, the government is sending in respected diplomat Carlos Sada to lead a fightback.

Mexico's new ambassador in Washington, Sada acknowledges his country has neglected its image across the border and aims to fix that with PR and media campaigns, and by lobbying prominent U.S. companies, lawmakers and civic leaders.

"We need to do a more thorough job so that people understand what (Mexico) contributes," he said after he was sworn in at Mexico's Senate on Thursday.

Sada's strategy includes underscoring Mexico's importance to the U.S. economy, although it centers on defending the rights of Mexican citizens in the United States and promoting Mexican culture.

That focus has fed doubts over whether the government is trying hard enough to win over its most important audience: American voters.

"It's vital to improve Mexico's image and protect our people, but that's not enough to change the hateful trend that Trump and other xenophobes before him have stirred up," said Gabriela Cuevas, an opposition lawmaker who chairs the Senate's foreign relations committee.

"They don't understand the extent of the damage Trump has done," she said, urging the government to be more aggressive in mobilizing powerful U.S. interests against Trump's attacks.

Claiming Mexico is "killing" the United States on trade, Trump has threatened to disrupt bilateral commerce worth some $500 billion a year, and promises to deport millions of undocumented migrants from Mexico and Central America.

To finance a border wall to keep migrants out, he has controversially proposed blocking billions of dollars in remittances sent home by Mexicans in the United States.

The measures would pose a serious threat to Mexico's economy, but for months Mexico's government disregarded Trump, hoping his candidacy would fizzle out.

"It's the ostrich policy: head in the sand," said Agustin Barrios Gomez, the head of Fundacion Imagen Mexico, a group dedicated to promoting Mexico's image abroad.

Mexican officials say U.S. politicians and officials urged it to keep a low profile to avoid aggravating tensions, and played down the real estate magnate's chances.

"The Republicans told us, 'We'll deal with Trump'," one senior Mexican government official said, speaking on condition of anonymity.

When Mexico eventually did respond, President Enrique Pena Nieto compared the brash billionaire' s campaign to the rise of Hitler and Benito Mussolini.

Mexican officials wince with embarrassment when reminded of the comment, arguing it was tactless and went too far.

EMPTY EMBASSY

As Trump railed against Mexico, the government should have made a concerted effort to remind key players in the United States that the two nations' economic interests are closely intertwined, diplomats and business leaders say.

But over a dozen serving and former senior Mexican officials and lawmakers consulted by Reuters said it did not.

"They haven't so far, but I do see the intention to do it again," said Jaime Serra, a former trade minister who headed Mexico's negotiation of the North American Free Trade Agreement (NAFTA) with the United States and Canada in the early 1990s.

Trump has not been the only one to criticize Mexico.

His Republican rival Ted Cruz also supports a border wall, and backs mass deportations of illegal immigrants.

Democratic hopeful Bernie Sanders has, like Trump, taken a protectionist line on jobs and says NAFTA was a mistake.

Mexico's cause was not helped by Pena Nieto leaving his diplomatic mission in Washington without an ambassador for six months just as Trump was warming up.

And his eventual choice surprised many: Miguel Basanez, an old friend who had never worked in the diplomatic service.

"It was a bad decision from the start," said a senior lawmaker inside Pena Nieto's ruling Institutional Revolutionary Party, or PRI. "They didn't grasp the size of the problem."

Basanez was cast aside this month, just seven months into the job.

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