This will take some brain power to comprehend, but here we go:
I have to say that John White can't be the Beast from Kessler's timeline.
Kessler speeds up the production of Ray Sphere, so Cole is younger than Kessler was when he got powers. Let's say that Cole is 25 when he gets his powers. Kessler would have been at least a few years older (because he got his powers, and then started a family), so let's say he was 30. The Beast doesn't attack Kessler for a few years, because it is shown that he has children, the oldest at least 4 years old, so when the Beast finishes his rampage (maybe a year-long campaign) Kessler is at least 34.
John White was in Empire City for a little while before Cole got his powers. In Kessler's timeline, John would have left the First Sons early (who knows, he may never have even infiltrated, because in Kessler's timeline, Alden would have been the leader), because there was nothing really happening (because Kessler wasn't speeding up development of the Ray Sphere). So John would have left Empire at least 8 years before the Beast attacked. Kessler never sets the Ray Sphere off a second time (of course not, why would he; he has family) so no one was absorbed by the Ray Sphere to go on to become the Beast.
It is more likely that an alternate timeline of evil Cole (assuming that Kessler is good future Cole) got the same power of time travel, then went back in time and became 'the Beast'. This is also reinforced by the fact that the Beast from Kessler's timeline can use Ice powers, as seen when Kessler shows Cole the future just before he dies. (The raven is picking the flesh of someone frozen in ice.)