I think the fingering in the score is good. Funnily enough I find the leaps more difficult than the thirds, but it might be because I don't have the piece in my fingers anymore. 135 bpm is quite fast but of course it is doable, my favourite recordings (Annie Fischer and RIchter) play that part at around 133 bpm.
As with most technical problems, I think you need to be relaxed, light and use economy of motion to succeed in this part. So watch out that you are not tensing your hands/fingers because it feels tricky or stressful. If the non-playing fingers are even a bit tense when you play a third, the next third will feel more difficult to play and it may cause you trouble attaining speed. Make sure it feels very smooth and comfortable when you practice, and not angular or forceful or anything like that. Lateral rather than vertical if that makes sense.
If the crossings are difficult, make sure that you allow your arm to be totally free to follow your hands/fingers to the new position on the keyboard, that you are not tensing/restricting your arm from moving as much and as quickly and as smoothly as needed. Sometimes if your arm is not free you can sort of end up tensing your fingers kind of stretching/reaching for the new position to compensate for your arm not getting your hand into position in time.
I know this is kinda vague and general but I hope it helps.