Providence the restaurant, or Providence the city? BYOB is legal in Providence RI. Plenty of Italian restaurants on Federal Hill. Did BYBO at Blue Grotto, can't remember what the corkage was but it wasn't anywhere near $30.
You might get better responses regarding the finer points of Providence's wine list on the Los Angeles board. Anyway, I would call the restaurant and ask them to fax or mail a copy of the wine list to me so you can have a look.