At least one blogger has said that BA gives its own F passengers AA Flagship First Dining at DFW. It's not on AA's website though.
Even though the AA website says what it says, AA has granted me access to Flagship First dining on BA F flights in both directions (i.e. connecting coming off a BA F flight and going onto a BA F flight; both times in MIA). Flagship First dining is really nice. Bottom line is it doesn't hurt to ask nicely with no expectations. There is no Flagship First dining at ORD b/c AA doesn't have Flagship First flights from Chicago. But DFW is possible.