* Paul buoyed after strong New Hampshire, Iowa showing

* Third party run could divide Republicans

* DeMint urges party to adopt some libertarian views

By Deborah Charles

COLUMBIA, S.C., Jan 12 (Reuters) - The Republican Party is faced with a dilemma: how to handle a popular, unorthodox presidential candidate who wants to do away with the Federal Reserve and end U.S. military presence overseas.

Ron Paul's libertarian philosophy is resonating with voters and senior Republicans say the party needs to show respect for him and grant some concessions to make sure he does not run as a third party candidate, and take his supporters along with him.

With a tough battle ahead to prevent Democratic President Barack Obama's re-election in November, Republican politicians and strategists say Texas Congressman Paul - who has spouted his anti-establishment views for decades without much success - must now be listened to, at least in part.

"In this election you can't afford to lose any voters," said Tony Fratto, a Republican consultant and former deputy spokesman for President George W. Bush.

Paul pulled in about a quarter of the vote in the New Hampshire Republican primary Tuesday after winning about 21 percent of the vote in Iowa.

While this was not enough to win, it proved the strength of his loyal following, much of it made up of young people.

"It's really important for whomever the Republican nominee is ... to treat the Ron Paul constituency with respect, to not be dismissive, to appreciate their energy and enthusiasm," said Fratto.

Paul will not likely have as strong a showing in South Carolina on Jan. 21 compared to Iowa and New Hamsphire, where voters relate more to Paul's libertarian, anti-establishment views. He is given a low chance of winning the nomination and many of his supporters may not transfer their votes to another Republican if Paul drops out.

What the party does not want is for Paul to decide to run as a third party candidate. That would divide some Republicans and attract independents wanting a change, making it tougher to defeat Obama in the Nov. 6 election.

A THIRD PARTY RUN BY PAUL COULD 'CREATE HAVOC'

Ed Rollins, a veteran Republician strategist who ran Mike Huckabee's 2008 presidential campaign and Michele Bachmann's for awhile in 2011, said Paul needs to know he is being heard.

"If for some reason he's not treated well and chooses to run as a third party candidate he would create havoc," said Rollins, who suggested the party have an establishment representative consult and liaise with Paul.

Senator Jim DeMint of South Carolina, a favorite of the conservative anti-Washington Tea Party movement, said he thinks the Republican Party should embrace more libertarian ideas.

"I think one of the things that have hurt the so-called conservative alternative is saying derogatory things about Ron Paul," DeMint said on Wednesday in an interview with conservative radio talk show host Laura Ingraham.

"I don't agree with him on everything, but he is right about the out-of-control and unaccountable Federal Reserve," said DeMint.

"If the other candidates miss the wisdom in what he's saying on monetary policy and limited government, then I think we will see it's to their detriment because the 20 percent or 25 percent or so who are supporting him are people that we need in the Republican Party."

Some of Paul's proposals on monetary policy could be included in the Republican platform, suggested some strategists who all agreed no other Republican would adopt the radical foreign policy views Paul espouses like practically doing away with U.S. involvement overseas and massive military cuts.

Paul may already be impacting some candidates' policies.

Mitt Romney, the front-runner who has won in Iowa and New Hamsphire, has promised deep spending cuts, smaller government and a new Federal Reserve chairman - proposals that seem designed to appeal to the radicalized conservative base of the party enamored of the gold-standard ideas promoted by Paul.

Paul, who is running for president for the third time, and his supporters insist he is a viable threat to the nomination but in the end the former obstetrician may just want to be taken seriously. Perhaps even win a coveted speaking role at the Republican National Convention.

Rollins said the 76-year-old Paul, who is not likely to pull out of the race any time soon, could end up a second place finisher to Romney in the race for the Republican nomination.

"If he's second then you have to give him a speaking role at the convention," Rollins said, though the Republican establishment might cringe at the thought of what Paul could say in a prime-time televised speech.

"I think he will be a force in the convention that the nominee has to go to and ask 'What do you need not to be a problem?'" said Cal Jillson, a political analyst at the Southern Methodist Univeristy in Texas.

If Paul is looking for concessions as a carrot to not launch a third-party bid, the eventual nominee could include him in studying the Federal Reserve or give him a role in choosing a new chairman, suggested Republican consultant Matt Mackowiak. (Additional reporting by Steve Holland in Greenville and Stella Dawson in Washington; Editing by Alistair Bell and Jackie Frank)