What could FDR do under World war II, to end the war earlier than it ended?

It seems like the pertinent historical question embedded within your question is: how difficult was it to win the Second World War?

Answer: Pretty difficult.

A full answer would be very multifaceted, even assuming we limit the question to the strategic agency of only the United States (given the focus on FDR). I'll just give a broad overview as a sort of introduction. First, the US industry had to gear up for war, which began before US entry into the war but was only completed in its latter half. The USA eventually produced over 100 carriers (a dozen or more being proper fleet carriers, the rest light and escort carriers) during the war. The USA produced tens of thousands of aircraft (an accomplishment made all the more notable because a large portion of them were massive and complicated 4-engine heavy bombers), as well as tens of thousands of tanks. Not to mention small arms, other naval vessels, ammunition, the Liberty ships, the atomic bomb, etc. Materiel is, however, only a small part of it. Without a working strategy, materiel will not prevail. Without functional logistics, even with a reasonable strategy, the materiel can't be gotten to where it needs to be. Without personnel, the materiel can't be used. The US military forces of 1945 were not the US military forces of 1941. All of this takes time to acquire, to employ, and to get right. And even then things will go wrong due to friction, enemy action, the weather, and so on.

Therefore, we must discuss strategy. When the USA joined the war in December 1941, the Axis held the initiative everywhere. Germany was deep in the Soviet Union; the Brits were failing to achieve decisive success in the Western Desert; Japan was overrunning Malaya, the Dutch East Indies, Burma, and the islands of the Central and Southern Pacific. 1942 was the year when Axis successes reached their greatest extent. Not for nothing did Winston Groom call 1942 the year that tried men's souls. It was only by the end of 1942 that any of the Allies and co-belligerents were in a position from which they could begin conducting serious offensive operations--but mostly only in peripheral theaters (except for the Soviet Union), and with relatively small amounts of men. And even then they had it hard. As late as February 1943, the Germans could route American forces at Kasserine Pass in Tunisia. There was no way the American army was ready for a cross-channel invasion. The British Army wholly lacked the power to embark on such a project on its own. The tyrannies of logistics meant that, after North Africa, the most effective place to continue operating was in the Mediterranean, therefore the Allies invaded Sicily and then Italy. In Italy they got bogged down by a combination of torturous terrain, bad weather, and a skilled German defense. And to think, the US and UK were only facing a fraction of the German army; most of the Wehrmacht was being ground up in the east. May/June 1944 was judged the earliest reasonable time to invade across the Channel, and it was done. Then came more fighting against an adversary who was very skilled tactically and operationally even though his strategy was in utter shambles. As it was, the Allies got lucky with some coups such as capturing the bridge at Remagen intact, or that by the end the Germans were barely resisting the Allies, preferring instead to be able to surrender to them instead of the Soviets.

The war against Japan was even more dominated by logistics. Getting the supplies necessary to support major fleets, not to mention some significant armies in the South Pacific and ultimately the Philippines, is hard. The Fleet Train is justly praised as an amazing achievement. Even so, Japan resisted on just about every island the Americans landed on, from Guadalcanal and Papua New Guinea to Tarawa, Peleliu, Iwo Jima, and Okinawa. As it is, MacArthur was pretty savvy and bypassed loads of fortified islands, leaving the Japanese garrisons there--tens if not hundreds of thousands of men all told--to wither on the vine, which took the Japanese entirely by surprise. The carriers of the Japanese Navy were decimated at Midway, but the Great Marianas Turkey Shoot, their last throw of the carrier dice, occurred only in 1944. The last great naval battle at Leyte Gulf took place in October of that year, after which the Japanese had to resort to kamikazes and the suicidal voyage of the Yamato because only then was it scraping the bottom of the barrel. Soon FDR dies, and it's left to Truman to make the ultimate US choice on shortening the war: dropping two atomic bombs.

To summarize: strategy is hard, and the Allies faced foes who were generally tactically and operationally competent (to say the least) even if strategically idiotic. Strategy is done as tactics, therefore despite their strategic inadequacy the tactical and operational prowess of the various Axis armed forces dragged the war on for quite a long time, despite the Allies doing everything they could to achieve victory as quickly as possible and some occasional luck. This was really just a very broad, introductory overview.