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UPDATE 1-Run or wait? Tokyo's Koike faces dilemma ahead of Oct. 22 poll

* Tokyo's Koike could risk backlash if resigns to run

* Koike again denies plan to run, speculation persists

* Democratic Party rump liberals form new party

* Voters split on hopes for "Party of Hope" (Adds public opinion poll, launch of new liberal party)

TOKYO, Oct 2 - Tokyo Governor Yuriko Koike has made no secret of her desire to be Japan's first female prime minister - she even named her pet terrier "Sori", Japanese for "premier".

But Koike, 65, whose fledgling Party of Hope poses a growing threat to Prime Minister Shinzo Abe's ruling bloc in an election this month, faces a tough choice: run now for a seat in parliament so she becomes eligible for the nation's top job, or wait and bet her party positions itself to win the next national poll.

Abe called the Oct. 22 poll in hopes his Liberal Democratic Party-led bloc could keep its majority in the lower house, where it now holds a two-thirds "super majority", but Koike's party has upended forecasts.

If Koike resigns as governor little more than a year after defying Abe's LDP to run successfully for that post, she would risk a backlash from voters. Waiting might let her best shot at the premiership slip through her fingers.

"If she thinks of the nation, it is important that she boldly announce her candidacy, present her ideas about important matters ... and debate policies head on," Abe's ally, Chief Cabinet Secretary Yoshihide Suga, told reporters on Monday.

Koike, a media-savvy former LDP member and defence minister, has said she would not resign as governor to run now, especially ahead of the 2020 Olympic Games, which Tokyo will host.

But her carefully phrased remarks have failed to kill speculation that she will run, perhaps announcing on Thursday, when the Tokyo Metropolitan assembly ends its session. Candidates must register on Oct. 10, when the campaign officially starts.

Koike is certain to be watching opinion polls as she weighs her decision.

A TV Asahi survey published on Monday showed 72 percent were negative about her potential candidacy. A separate NHK poll showed voters were split on Koike's new party, with about 47 percent having hopes for it versus the same percentage who had little or no expectations.

Further complicating the election outlook, a senior Democratic Party executive, Yukio Edano, announced on Monday the formation of a new Constitutional Democratic Party - with fellow liberals.

"NEXT, NEXT" ELECTION?

Koike's close ally, Masaru Wakasa, on Sunday said it was not essential that she run this time around.

"If there is a firm prospect of a change in government, party chief Koike might run for parliament, but if we can achieve a change in the 'next, next' election, she doesn't have to run now," Wakasa said during a debate on NHK.

Koike, however, has already once blindsided Wakasa, who led efforts to form the new party, by announcing - on the day Abe announced the snap poll, that she would lead the Party of Hope herself, not entrust it to allies.

On Monday, she told Jiji news agency her party aimed to run more than the 233 candidates needed to take a majority in the 465-seat lower house. "If you don't buy a lottery ticket, you can't win," she said.

But she again denied she would seek a seat in parliament now. "I was in national politics for 24 years and I have no intention to go back to a place where you can achieve nothing despite spending so much time," she said.

A former TV announcer who speaks Arabic and English and peppers her speeches with foreign words, Koike wants to build a "pro-reform, conservative" party to compete with the LDP, in part by cannibalising the failed main opposition Democratic Party and luring smaller parties to her side.

Her cool, polished demeanour belies a steeliness beneath.

"She can take a punch with a smile and sharpen her knives at the same time," said Jesper Koll, head of equity fund Wisdom Tree, who has followed Japanese politics for years.

POPULAR POLICIES

Like former prime minister Junichiro Koizumi, one of her past mentors, Koike promises to "break free of vested interests", a slogan that resonates with voters looking for an alternative to the LDP - although critics question just how different that alternative would be from Abe's LDP.

To differentiate her party from the LDP, Koike has adopted popular policies, such as an end to atomic power amid public safety worries after the 2011 Fukushima nuclear crisis, and proposes to freeze a planned sales tax hike from 2019.

But she also backs Abe's push to expand the role of the military overseas and his goal of revising the post-war constitution - although recently she has avoided focusing on the divisive issue of amending its pacifist Article 9.

She has also said in the past that Japan should consider having nuclear weapons, breaking a taboo in the only country to suffer an atomic attack, and, like Abe, has visited Yasukuni Shrine for war dead, seen in China and South Korea as a symbol of Japan's past militarism.

Koike entered politics in a small reformist party and migrated through several other groups before joining the LDP and in 2008 became the first woman to run for head of the conservative party. She came in a distant third. (Reporting by Linda Sieg, with additional reporting by Kaori Kaneko and Kiyoshi Takenaka; Editing by Ian Geoghegan)