If Pirate Joes is a Trademark violation, then every single small business that includes a persons name is a trademark violation. Regardless, settle a name change, and done. My point is there is zero need to be litigious here, and there is no logical legal justification.
Trader Joeys, clear trademark violation. Pirate Joes, no fucking way. Not in this country.
If the store named "(thing) Joe's" builds its entire business on "we sell Trader Joe's products", there is a very reasonable case that they're confusing (possibly deliberately confusing) the public about whether they are or are not an actual Trader Joe's. Which... brings trademark law into play.
So, like I said. Maybe it's time for you to go back to law school.
It is not plausible that "Pirate Joe's" is confusing. You can't simply discard "(thing)" and focus on "Joe's", trademarks are taken with their entire context.