My own idea of a plausible third party candidate who might be (marginally) stronger than Lemke was in OTL: Have Bronson Cutting survive his plane crash in 1935. It's surprisingly easy:
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It turns out that Cutting didn't have to die.
By that I don't mean simply that he might not have boarded the fatal
flight, but that even if he did board it and even if it did crash as in
OTL, he didn't have to die--eight of the thirteen people on board the
plane survived, and of them five had only minor leg and rib fractures.
His fatal mistake was probably changing seats:
"All fatalities were seated in the extreme front of the craft. From the
way the plane hit the ground, passengers in the right front must have been
killed instantly, as was the copilot. Cutting was assigned seats 9 and 11
in the rear of the plane where no one was seriously injured. Where Cutting
was sitting at the time of the accident was unclear. His younger sister,
who compiled a memorandum on the accident explained, 'It is practically
impossible to account for the injuries sustained unless he had been
sitting right behind the pilots.'
"Why did Cutting change his seat? In the single seat at the back sat a
woman with a baby whose crying kept him awake. Thus it seems probable
that Cutting moved to a vacant seat in the front of the plane in order to
sleep. When the pilot light, a sign to fasten seat belts, came on shortly
before the crash, Cutting did not comply, indicating that like most
passengers he was probably asleep. His injuries were confirmed by a
doctor at the Samaritan Hospital at Macon, Missouri, where the injured and
dead were taken. Limited to his head, they indicated that death was
instantaneous.
"Joe Breeson, one of the first persons to reach the scene, saw eight
people lying on the ground. With the assistance of a couple of farmers, he
pulled away the debris and entered the cabin. In the plane, there were
three people one of whom was Cutting. He was dead on the floor up near the
front of the cabin. His head was crushed; his body was wedged in between
two seats. Only after considerable prying did rescuers finally release
his body. One of the survivors believed 'that Senator Cutting died
without awakening or knowing what had happened,' reinforcing the
conclusions of the doctor who viewed his body at Samaritan Hospital."
Richard Lowitt, *Bronson M. Cutting: Progressive Politician* (Albuquerque:
University of New Mexico Press, 1992), pp. 308-9.
https://groups.google.com/d/msg/soc.history.what-if/_5sf-smj9Nk/UDqaCGb71rcJ
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Something tells me that having survived the crash, Cutting might not be in a very good mood about FDR...