Understood. But Petain signed the Armistice June 22, 1940-- before the July convention. So the ceasefire with Germany (as stated in the OP) must be a British ceasefire (even though no one in England was willing to negotiate with Hitler at that point). But even if one plausibly changed the British policy of negotiating a truce with Hitler (the rocky shoals on which many contributors have perished), would these ceasefire negotiations be kept secret from the US?
I agree that the British negotiating with Hitler is extremely unlikely, but that is the conceit of the OP's timeline, so we just have to roll with it.
The US leadership was very concerned that there would be a very real and successful invasion of the UK in the coming months and that particular danger formed the publicly-stated basis for FDR breaking the 3rd term norm. Of course, the proposed TL could have Roosevelt privately planning to run in 1940 and simply needing the appropriate pretext to break with tradition. Then Wallace is on the ticket as Roosevelt's VP for the convention. But now there is no war waging in Europe on US election day and either the UK-Nazi truce is a done deal but if not, the negotiations are commonly known in the US. Pearl Harbor hasn't happened and the only place Japan is running wild during 1940 is in China. So now Roosevelt would be running for a 3rd term (which had also been a 3rd rail in American politics) with no reason for violating tradition (there's a greatly reduced threat of war and the Depression is increasingly in the country's rearview mirror). But let's assume FDR concocts something that US voters accept and re-elect him.
The Nazis dominating Europe isn't enough reason? Do you honestly think the American people are going to think Hitler is less of a threat once he's conquered France and beaten Britain? The American people (and leadership) were absolutely terrified by the fall of France. Just look at how the US reacted to that IOTL with the very prompt passage of the Two Ocean Navy Act, a massive increase in defense spending. That act authorized a staggering number of new ships and planes.
(The above list per Wikipedia).
But what's really interesting about the Two Ocean Navy Act is the process by which it was passed. Admiral Stark asked Congress for four billion dollars to expand the fleet, and the House of Representatives more than doubled his request, approving an 8.55 billion appropriation, after less than a hour's debate, unanimously. Unanimously passing a bill with more than twice the funding level the military asked for after only a hour's debate is the mark of an absolutely terrified country, and the US anxiety level is certainly not going to be reduced by the news that the British are negotiating with Hitler. (If anything that news is going to hyper-charge US anxiety since that means there's now nothing standing between Hitler and the United States.)
Thus FDR has all the justification he could ever need if he wants to seek a third term because the Nazi threat looks even greater than IOTL and the American people are (understandably) going to be far more afraid here than they were IOTL.
I'm not yet heard anything compelling, or a plausible scenario, where Wallace has a shelf-life past the 1944 convention. By 1944, (per the OP) there is a truce with the UK that's held up, so the US isn't at war in Western Europe. Moreover, Nazi Germany is now involved in resource intensive war or occupation involving the Soviets. That means zero risk of Hitler yelling "fooled you" and opening a second front with another double-cross. Therefore, to have a "wartime presidency" for Wallace, the OP needs a Japan that decided to go ahead and take on a US/UK/Commonwealth alliance that in the proposed TL is not pre-occupied in Europe, the Middle East and North Africa, and then, Japan would need to survive nearly as long in that conflict as they did OTL And can anyone trust Wallace not to advocate (or suggest) that the US should declare on the Nazis to help the Soviets!?!
Imperial Japan was not exactly known for it's rational foreign policy decision making. IOTL when they were already bogged down in China they still decided to take on an (at the time undistracted) United States, a country with 16 times their industrial capability and to also fight the British Empire while they were at it. If faced with a choice between taking on nation's that are significantly more powerful than them or withdrawing from China, they are going to chose the former no matter how insane it is. We know this because that's exactly what they did IOTL.
Furthermore, the necessary conditions for a Pacific War are all still there since if Roosevelt is reelected in 1940, he will almost certainly extend significant military aid to China in 1941.
IOTL, the US in 1941 agreed to provide the resources necessary to provide the Nationalist Chinese with 500 planes and to equip 30 Chinese divisions to US standards. (Per "Stillwell's Mission to China" from the US Army Center of Military History, the Chinese were specifically requesting 30 battalions of 75mm pack howitzers with 2000 shells for each gun, 30 battalions of 37mm anti-tank guns with 1,500 shells for each gun, 30 battalions of 105mm and 8 battalions of 155mm howitzers, 15000 7.92 mm machine guns with 500,000,000 rounds of ammo, 360 light tanks and 400 scout cars which the US believed it could deliver by mid 1942.) I don't think a lack of war in Europe would prevent the US from agreeing to support the Chinese in 1941 since FDR is still sympathetic to China, the China Lobby is still a powerful voice in US politics, and Americans will still fear Japanese expansionism regardless of what is happening in Europe. (Plus with no war in Europe, it's actually easier for the US to justify sending military supplies to China since that equipment is not needed for Britain.)
Japan can't win in China if the Chinese get that level of aid from the US, and the Japanese will know that. Thus they either have to prevent the US from shipping that aid to China which means invading Indochina and Burma (even if that means war with the US and UK) or accept the loss of China. And we know that when faced with that choice the Japanese leadership will chose war. (They pretty much have to because agreeing to withdraw from China means them getting assassinated by their own junior officers.)
As for Japan surviving as long as they did IOTL, that's very likely as well. Warships (and more importantly a fleet train) can only be built so fast, and thus it is going to take several years for the US to build up the necessary fleet and infrastructure to drive across the Pacific even if there is no war in Europe. Thus the war will almost certainly still be going in November 1944.)
As for Wallace wanting war with the Nazis, even if Wallace wanted to do that, what would it matter? Wallace would need congressional approval to declare war (unless Hitler obliges him by declaring war first), and while Wallace may not have been the greatest of politicians, I think he was smart enough to know that Congress wasn't going to agree to go to war with the Nazis just to pull Stalin's chestnuts out of the fire.
The OP has Roosevelt dying during his 3rd term. Is this before or after Roosevelt makes it abundantly clear that Wallace has lost his confidence? I believe that was circa 1943.
In 1944 Roosevelt was still willing to write a letter stating he would personally vote for Wallace for vice president.
And even if Roosevelt did lose confidence in Wallace, why would that matter once Wallace is the incumbent president with all the prestige, power, and patronage that comes from that position? And especially when Wallace is presiding over an apparently successful war against Japan? (The Pacific War won't be over by November 1944, but it should be very clear the US is winning by that point.) The American people aren't going to want to throw out a president who is successful leading them through a war, and the Democratic power brokers would know that.
Lastly, why would Wallace's domestic policy have any success passing in an increasingly hostile Congress. Even FDR had problems with his favored domestic legislation post-1940. Congress (the home of many Wallace's intra-party enemies) should be an even bigger problem for a Wallace agenda.
Agreed but why is that relevant? I don't recall anyone saying that Wallace would be successful in enacting much legislatively.
My humble opinion-- Truman or Gardner or someone usurps Wallace at the 1944 convention and the OTL is restored with minimal butterflies. As such Wallace has no impact on the 1960 presidential race and is mainly viewed as a caretaker president.
In the unlikely event that the Democrats are able to topple a wartime president presiding over a successful war, it would still require a nasty nomination fight that would split the Democratic Party in two. Thus the result of such an event would almost certainly be President Thomas Dewey come January 20, 1945.