No issues at domestic lounges at any time with any of my supps. Only issue I had when I was a supp charge holder was in HKG Centurion lounge - on entry they scanned the card, questioned me on "who's account this was" and after explaining (that I was good friends with the account holder), they let me in.
I think supplementary card holders of the platinum charge officially have access to the centurion lounge.