When I use Naganadel GX's "Injection GX" attack, am I allowed to mix up my opponent's prize cards so they don't know which card I added to their prizes?
No, you may not. When Injection GX adds a card from the discard pile you must not disturb the placement of any prize cards the opponent still has in play.
Source: Unified Minds FAQ (2019-07-19), TPCi Rules Team (2019-08-01)

Relevant cards

Note: All applicable cards may not be listed below.
SM11_160
SM11_160
SM11_230
SM11_230
SM11_249
SM11_249