I repair a lot of iPads and I have never seen, nor can I logically explain, why the iPad wouldn't charge or mount even if the LCD is bad. I can't imagine the LCD can corrupt the logic board into not charging or mounting.
Even in the case of bad LCDs, I could still pull it up on the computer and charge it. It sounds like you did your due diligence in what you tried so I can only think that trying a different LCD (one that you feel comfortable giving to the customer) would be possible to get it to work.
Have you tried swapping out the LCD cable on that ugly LCD (which works) with the newer LCD? I still cant see that being the issue but it would be worth a try before you shell out for another LCD.