What is the best way to shuffle a large deck without damaging the cards?
With the implied restriction on time combined with the explicit restriction to keep the cards pristine, I'd suggest using extra space to shuffle.  Here's what I do with large decks of cards:


Divide the large deck into several smaller piles.  For 80 cards, I'd make about 8 piles of 7-15 cards
Select two piles at random and combine them using a hindu shuffle (unlike in the video, when I do it, I do it by grasping the wider-side rather than the narrower side of the cards for better control of larger decks.
After a few shuffles, grab another random pile and add it to the deck
Continue until all decks have been added.
Continue with several more Hindu shuffles until you're satisfied.


If you cannot keep all cards in your hand, recombine/shuffle half the cards, then recombine/shuffle the other half, put the two halves together (just stacking them) into 1 full deck again.  Finally, repeat the entire process two or three times (this is only necessary if you had to do a simple stacking).