Holy cow. I just realized Puzzlemad's algorithm for the corner swap makes things a million times easier....
but he's just flipped my world upside with that simple change. My wall of text was completely unnecessary and solving big cuboids will be SOOO much faster now. Wow. I feel stupid.
Wow! I didn't even flip my wife's world upside down when I married her!
The little things make all the difference! I still like your parity fix!