I have mostly reconciled myself with this one but I thought I'd ask the question anyway.
Discard this card to choose a character at your location and add 1d10 to that character's check to defeat an ally that has the Pirate trait.
Here comes the Captain ally (check to acquire: Constitution/Fortitude 4 then Charisma/Diplomacy 6). Rum Bottle doesn't apply to both checks; you have to choose one, or the other (unless you or a pal at the same location have 2 Rum Bottles between you and play one for each check). Right?
Right. You'd play it during "Attempt the Check", specifically during "play cards and use powers that affect the check." And just like lots of other thing played then, they don't have a lasting effect beyond that one check, so when you get to "Attempt the Next Check" it is no longer doing anything.
I'd also suspect that, if it was intended to effect multiple checks, it would be displayed instead of discarded.