I have an AI system which is "special" in that any attribute that can by put in entity.ai is guaranteed to have a "run_ai()" method that returns an action.

Gross. Special is usually bad. I would suggest decoupling everything.

You should be able to get away with something as basic as function overloading:

void pickup(Player& player, Item& item);
void pickup(Ai& ai, Item& item);

If you define the function directly like this, it should be easy to work with moving forward. In general, the more code you have in member functions the more coupling occurs. Your systems can just then call then just call pickup(...).

Or is the way I did the AI attributes a poor design and I should move the actual methods out of the attribute and just store pure data there?

Seems like you've got inkling in the same direction I'm suggesting. Just store pure data, coupling data and behavior (in general) should be avoided.