The effect itself sounds sort of like the Lovetone Big Cheese, which can achieve a pretty good range of sounds so it’s a good bet.
It doesn’t really fade between clean and fuzzy, but just jumps to lightly fuzzy at 2:16. Then it jumps to the sustained very fuzzy notes at 2:25, then back to slightly fuzzy chords at 2:32. This “pattern” basically repeats until the end of the song.
Considering that it’s a recording, each part was probably recorded separately. And indeed, if you listen at 2:37-2:42, you can hear both the sustained very fuzzy note and the chords at the same time.
However, it is possible to pull off this kind of thing in one go without too much difficulty. As you probably know, fuzzes and distortion pedals “clean up” when the input level is decreased. Therefore, one option is to be insanely fast with the volume knob, very quickly lowering and raising it between each part. That isn’t all that practical, though, and sounds like it’d be a little tedious to do. An easier option would be to put a volume pedal before the fuzz, and adjust the input level that way. This option would be pretty easy to implement and could yield the exact same result.
Also, the fuzzy guitar seems to come in at the very beginning of the song, I think it might be responsible for those fuzzy popping sounds.