What bothered FDR was not the prospect of Long actually winning but of his getting four million votes--about one-tenth of the electorate. That was the figure given by a secret poll commissioned by Jim Farley in early 1935. Of course we now know that even had Long gotten that many votes it couldn't possibly have changed the outcome, FDR having defeated Landon by over eleven million votes. (For that matter, four million votes for Long couldn't even have prevented FDR's victory over Willkie in 1940, even if we make the unrealistic assumption that Long's votes would all come at the expense of FDR.) But FDR had no way of knowing that in 1935; many people expected the 1936 race to be close. And FDR even suspected that Long, despite his rhetoric about redistribution of wealth, was in a secret alliance with the conservative Liberty League to bring about a Republican victory in 1936.
In any event, it is very doubtful that Long could even win those four million votes. In general, third party candidates do worse in actual elections than in polls, especially polls taken several months before the election. (The "novelty factor" wears off and the "it's a wasted vote because he can't win" argument becomes more widely accepted as Election Day approaches.) Furthermore, in this case, there is an additional reason to expect Long's support to decline--the economy improved considerably between the time Farley's poll was taken in 1935 and Election Day of 1936.
Anyway, after Long would lose in 1936, he would probably either go to jail or be assassinated before 1940. But if he did survive and would run again, he would lose again, and as already mentioned would not even take enough votes from FDR to give the election to the Republicans.
I would say that to have FDR defeated in 1936, with or without Long in the race, you would have to have the US economy do much worse than it did in OTL. How many people realize that in that year "real GDP grew 13.1 percent and the unemployment rate fell 4.4 percentage points..."?
http://behl.berkeley.edu/files/2013/02/WP2013-06_Hausman.pdf And it would be hard to maintain a successful populist campaign against FDR--a man who campaigned against "economic royalists" and who was characterized by a southern millworker as "the first man in the White House to understand that my boss is a son of a bitch."
https://books.google.com/books?id=EFb4jFcfDVQC&pg=PA46