The question I have is around whether it is deemed 'incorrect' to play some of the left hand in the right hand. Although I started learning it as scored, I'm finding I can get much smoother results by somewhat cheating.
I don't think this is cheating at all, and it actually is common practice, as I was told by my former Moscow Conservatory trained teacher. Use whatever hand works best for you - as long as you don't hear any of it happening of course. In polyphonic music, taking some note(s) with the other hand if often a bare necessity.
My organ teacher is also in favour of hands helping out each other. He had the amusing metaphor of two workmen, one of whom is digging a hole and the other is standing next to it, watching and smoking a cigarette.