So long as it is a limited war with limited goals, this is not so very difficult. Here is my quick-and-dirty timeline.
Prelude
In 1926, FDR dies as a result of illness.
Herbert Hoover decides not to run again in 1932. After a close but lackluster election, Charles G. Dawes becomes President of the United States. He remains unpopular in the Senate, and has to find a new coalition on every issue. The Great Depression reaches its lowest point. It is clear that something new must be done, but what remains unclear.
The Road to War
In Shanghai, nationalism and anti-foreigner sentiment is rising. In January, a Japanese-owned factory in the Japanese Settlement is burned down, allegedly by a Chinese mob. The Chinese Municipal Police engage in heavy-handed anti-riot patrols. One policeman and several civilians are killed in the scuffle between police and Chinese rioters. Following the disturbances, Japan sends 7,000 troops into Shanghai to protect its interests. Naturally, this merely inflames tensions further.
The Chinese (KMT) 19th Route Army marches to the edge of the city, as a show of force against the Japanese deployment. The Chinese business community, who depends on international trade for their livelihood, consider the 19th Route Army to be little more than bandits--almost as much of a danger as the Japanese army. They raise a large sum of money, and pay the army to leave. The KMT forces retreat, but stay within a few days march of tne city. For now, there will be no war.
Japan chooses not to withdraw her troops, citing continuing unrest. A tense truce--more a simple lack of open war than peace--holds for a few months. However, on May 26, a bomb planted by Korean terrorist and independence activist Yoon Bong-Gil kills General Yoshinori Shirakawa, commander of the Japanese forces in the area. Yoon escapes, taking public credit for the attack in the media. Most people in Japan assume the Chinese are hiding him. The Japanese government demands that the Shanghai Municipal Government publicly condemn the attack, pay reperations, and turn over the terrorist. The first two demands are swiftly met, but the government cannot turn over a man it does not have. Japan invades Shanghai.
The Japanese forces in the area meet unexpectly stiff resistence. The 19th Route Army returns to the city to fight, surprising Chinese and Japanese alike. Japan expands its forces from 7,000 troops to over 90,000. The Japanese Navy also launches the first carrier air strike ever outside of Europe. Still, Japanese progress is slow. Chinese forces are dug in in a dense urban environment, and they make Japan pay in blood for every inch. Japan moves over 80 ships and 300 aircraft to the area to support their ground forces.
One of these aircraft makes a simple mistake, but one with big consequences. A Japanese Susie dive bomber, sent to attack Chinese strong points inside a building, accidentally drops a bomb on the American Embassy grounds by mistake. Two Marines on guard duty are killed. Japan swiftly apologies for the error, assuming the incident will blow over quickly. It does not.
President Dawes is facing increasing unpopularity at home. He is stymied in the legislature, and unable to pass any new laws of import. Increasingly, he is seen as a weak figure incapable of true leadership. To him, the incident in the Far East is a godsend. A war gives him just the chance he needs to display leadership. He gives a passionate speech in the House, harping on how the Japanese has attacked American soil and killed American men in uniform. He implies that this was done on purpose, and that the Japanese want to take over all of the international settlements in the city. The House, confident of a short, victorious war, signs off on Dawes' request. The US has declared war on Japan.
A Short, Victorious Little War
The US assembles a grand fleet and sails west from San Diego. They expect the Japanese navy to steam east into the Pacific, or perhaps to attack the Phillipines. Instead, the Japanese withdraw all their cruisers to the Home Islands for an emergency refit. This seems to indicate that the IJN is unready. It bolsters the confidence of the US sailors.
Due to the refits going on in Japan, the US ships are able to make it all the way to the Phillipines without encountering any enemy ships. The refuel in Manilla Bay, and sail for Japan. However, the Japanese fleet is waiting for them in the Surigao Strait. Sighting the enemy at long last, the US forms a battle line and rushes towards the enemy. The Japanese fleet, however, has a surprise for them. Long before they get into gun range, the ships begin to explode. The Japanese ships returned to port to pick up their brand new toy: the Type 95 "Long Lance" torpedo. The torpedo is a shock to the Ameircans, whose own torpedos have only a third of the range. In fact, many captains believe that the ships must have been mined. Others, who understand that it was torpedos, believe that they must have come from submarines. No one can fathom torpedos with such range.
One by one, the Japanese cruisers fire their torpedos, turn, and retreat. In the end, they launch over 500 torpedos at the enemy in the space of only a few minutes. The Americans do not have time to react. Over 17% of their fleet is sunk outright. An additional 25% are damaged. The Japanese fleet leaves the seen. The smoke in the sky and burning oil on the water tells the story of the carnage they have wrought. The American ships do not persue. Instead, they spend the rest of the day picking up survivors.
That night, the Japanese return. The Americans have not practiced for a night battle. The IJN has. The results show. While Japan loses a great many ships, among them the Mutsu and the Fuso, the American navy is destroyed as a fighting force. Again and again, the Japanese ships are able to force the US navy into small groups, and then attack in greater numbers. The next day, only 1/3 of the orginal fleet returns to Manilla Bay. It is in Manilla Bay that they will spend the rest of the short war. Too small a force to engage the enemy, they are reduced to simply attempting to stop and enemy landing.
The Japanese try just such a landing the next day. Just over 5,000 troops come ashore in Bataan. They are decimated, and repulsed with heavy losses. In the grand scheme of things, it makes little difference whether or not the invasion is successful. The Japanese have shown they can land troops at will, and the Americans are now too weak to prevent it.
President Dawes is broken by these events. He has no choice but to go before Congress to ask for huge emergency funding to build new ships. In this climate of economic hardship, the Republicans refuse. One Senator openly wonders why thousands of American boys are dying because of a war between "Chinks and Japs". The most Dawes can get is a promise to rebuild the Navy slowly, at a rate of about one capital ship a year.
He has no choice but to sue for peace.
The Aftermath
Japan agrees to peace. They have lost many ships, and it will take years to replace them. In addition, many of their troops are still tied down in Shanghai, alough it is now clear that their victory there is a matter of when, not if. The eventual peace treaty ends up being little more than a status quo ante bellum. Japan is in no position to ask for reperations (angering many nationalists at home). They also ask for Gaum--an American possession quite close to the Japanese Pacific Mandates--but the US steadfastly refuses. In the end, Japan receieves only a promise that the US will not base capital ships in the Phillipines, and the American concessions in China.
The political aftermath of the war in both countries is perhaps more interesting than the war itself. In Japan, the militarists continue to gain in political power. However, now instead of the radicalism of the Army, it is the dreams of the Navy that comes to the fore. The difficulty the Army had in subduing one city in backwards China, compared with the glorious naval victory over a larger foe, tips the scales. (Of course, it is conveniently forgotten that the Army only entered Shanghai when the naval troops (SNLF) proved unable to do the job). Dreams of expansion from Manchuria into China are put on hold. Now, the Navy turns its sights to the South. The Admirals also now that a naval war may be a long one. It will be very costly to replace the lost ships. Therefore, their first priority will be to expand the industrial base.
In America, the political lessons are far different. An American humilation, in a war that many Americans didn't want in the first place, greatly stregthens American isolationism. The US no longer has the spirit for grand adventures. Instead, it simply licks its (economic) wounds, and slowly rebuilds. It is not until 1938 that the Navy returns to its prewar size. Privately, however, some Admirals say that the war might even have been a blessing in disguise. The new ships are far more modern--sometimes radically so--than the ones they replaced...