[quote]he opened the game, activated crossplay[/quote] If he got past the title screen - the passes were applied to their PS characters. Have them go here: https://help.bungie.net/hc/en-us/articles/360049497351-Contact-Form-Missing-Eververse-Item-or-Add-On-Items and fill in the form linked at the bottom. Whether Bungie will apply the passes/dungeon key is done at the reps discretion but they've been fairly good at applying passes in cases like this. Tickets are answered in the order they are received - during business hours monday->friday, they wont be handled overnight/over the weekend. They are fairly quick in situations like this to reply but they are a small team.
Is he using the same username or is he game sharing?