They can't afford not to. In particular, this is an absolutely golden opportunity for Germany to cut off American trade with Britain, which even in the absence of cash and carry and Lend-Lease is still a big deal for food and such stuff. Especially if the President is swearing up down and sideways that he's not going to send out the Navy and the Coast Guard to get in their way.
Germany stayed out of the Western Atlantic in OTL to avoid provoking the US; FDR sent US warships as far east as he could to assist Britain and possibly provoke an incident. Wheeler will keep US warships west of Greenland (to protect US shipping in our own waters), and will state flatly that any American merchant ships sailing to Britain do so at their own risk.
Why should Germany break that up?
And will the Japanese buy that? I don't think they will, to be honest. Why would they? From their perspective, at any moment the President could change his mind and send the Americans back in, and if the Japanese have already move past the Philippines on the grounds that they're neutralized and harmless they'll be caught flat-footed and with a hostile navy and army sitting astride key supply lines.
Right. Because the US could
easily move, say, 200,000 troops, 1,000 aircraft, and 50 warships 8,000 km from Hawaii to the Philippines, right through Japanese-held Micronesia, without the Japanese even noticing, much less interfering.
So they really have no reason whatsoever to believe the President...
Other than his long history of public statements opposing any US involvement in overseas wars.
...and plenty of reasons to seize the Philippines and Guam anyway...
Thereby guaranteeing a full-on war with the US and making the President's position moot.
In any case, I seriously doubt Burton K. Wheeler would actually be nominated by the Democratic Party in 1940, even in Roosevelt's absence. There's a reason Wendell Wilkie was chosen by the Republicans over Taft, after all.
Do you think it was just anti-isolationism? Wrong. It may surprise you, but most Americans were more concerned with domestic issues than foreign policy. Willkie was famous as an eloquent opponent of the New Deal. and his backers considered him more electable than Taft.
So more likely yet would be a President that willingly cooperates with Congress...just like Roosevelt.
FDR didn't "cooperate with Congress"; he led the way, with proposals such as Lend-Lease, which Congress approved, and with executive orders and directives to the armed forces for assistance to the Allies.
Wheeler would fail to cooperate with Congress only in the very unlikely event that Congress chose to declare war against the opposition of the President.