* Abe aims at extended run as PM
* Scandal over veterinary school hurts his ratings
* Capital's metropolitan assembly election seen as bellwether
* Future of post-war constitution key national issue
TOKYO, June 22 (Reuters) - Just a few months ago, Prime Minister Shinzo Abe looked to be cruising to a third term that would make him Japan's longest serving leader and put him on track to achieving his dream of revising its post-war, pacifist constitution.
But suspicions he may have helped a friend get favoured treatment for a business, then rammed legislation through parliament to close the session and end debate over the issue, have led to a sharp slump in support.
A metropolitan assembly election in Tokyo on July 2, campaigning for which starts on Friday, could give clues about how stable his administration really is - a key concern of global investors.
"Things are unravelling fast for Abe and his inner circle," said Gerry Curtis, a professor emeritus at Columbia University. "I still put my money on Abe surviving and getting a third term, but I am willing to wager much less than I would have put on the table a week ago."
At the core of Abe's troubles are concerns he may have intervened to help Kake Gakuen (Kake Educational Institution), whose director, Kotaro Kake, is a friend, win approval for a veterinary school in a special economic zone.
The government has not granted such an approval in decades due to perceived glut of veterinarians.
Abe has repeatedly denied doing Kake any favours.
Potentially more troublesome than the suspicions themselves is the impression among many voters that Abe and his aides, unchallenged and arrogant after more than four years in office, tried to suppress the scandal, in part by smearing an ex-official who went public with allegations and by rushing a contentious bill through parliament to close off debate.
"Chief Cabinet Secretary (Yoshihide) Suga's haughty attitude at his news conference and his attack on the former official personally were a mistake," said one Liberal Democratic Party lawmaker.
"That sort of condescending attitude makes people angry."
Abe, who resigned abruptly after a troubled 2006-2007 tenure as premier, can seek another three-year consecutive term as LDP leader, and hence premier, from September 2018.
No general election need be held until late 2018, but Tokyo voters have a chance to express their views soon.
On the surface, the Tokyo Metropolitan assembly election is a referendum on Governor Yuriko Koike, a former defence minister now challenging the old-boy network dominated by the LDP.
Koike, sometimes floated as a potential successor to Abe after the 2020 Tokyo Olympics, hopes her "Tokyo Residents First" party and its allies will take a majority in the 127-seat local legislature. Among her allies is the Komeito party, Abe's national coalition partner.
The capital's poll is often a bellwether for national trends.
"If the LDP loses half of its seats in the July Tokyo election, there will be worries about instability of the Abe administration," said Tomoaki Fujii, head of the investment research division at Akatsuki Securities Inc.
Koike may have lost some of her lustre lately after a year in office and delays in deciding whether the move of the world-famous Tsukiji fish market would go ahead.
The main opposition Democratic Party, meanwhile, is struggling with single digit support rates.
But the LDP is worried.
"Past Tokyo assembly elections have definitely affected national politics," Hakubun Shimomura, head of the LDP's Tokyo chapter and an Abe ally, told reporters this week.
The prime minister's ratings tumbled in surveys released this week, sliding 10 points to 36 percent in a Mainichi newspaper survey and 12 points to 49 percent in one by the Yomiuri newspaper.
Abe has pledged to regain voter trust, but his ruling bloc is resisting opposition calls to have committees address the Kake Gakuen reports while parliament is closed.
Opposition parties can demand an emergency session but the cabinet has ignored such demands in the past. An extra session of parliament is expected in the autumn.
Last week, under public pressure, the education ministry produced documents that opposition parties argued proved Abe was involved in the approval of Kake Gakuen's school.
The ministry had earlier said the documents could not be found in an internal probe, and chief cabinet secretary Suga said the idea that such documents existed was "dubious".
On Tuesday, the ministry also released an internal memo suggesting Deputy Chief Cabinet Secretary Koichi Hagiuda, a close Abe ally, was involved. Hagiuda has denied he had received or conveyed instructions from the premier.
Abe's support rates have dipped and then rebounded in the past. If they remain limp or worsen, LDP rivals would be encouraged to mount a challenge, and revising the constitution - a politically divisive step - would be tough.
The U.S.-drafted constitution has never been amended since it was adopted after World War Two. Conservatives are keen to revise what they see as a humiliating symbol of Japan's defeat, but others see the charter as the foundation for the country's post-war democracy and peace, and oppose altering it. (Additional reporting by Ami Miyazaki and Marika Tsuji; Editing by Sonya Hepinstall)