I'm playing with the distribution as written - I also have a pretty large reach though, so my left hand never felt strained...
It is a super awkward piece, and there's a general opinion on this board that it's "easy"... ignore that for sure! Take your time with fingering, but ya try little bits at a faster pace to make sure it's actually executable.
So for the right hand I'm taking the G# with 5, then playing the next key with 4... and always over top the left hand. I'm pretty sure that when I played it, I was able to avoid crossing for the right hand; the edition I have has pretty clear articulation marked, so some of the big leaps are not connected with a slur... I don't remember 100%, but I think I kind of 'leapt' a bit going up to the high C#. It also ensures that the quarter beat for the first sixteenth is sustained not just by the pedal... maybe try practicing without any pedal here, to see how well the marked articulation works with the fingering you're using.
I'm in a bit of a rush so this answer may not be super coherent. But let me know if you have other questions. I'd be happy to [try to] help!