I got around the same problem by adding a line of code to the setup of the SQLServerBulkCSVFileRecord object

SQLServerBulkCSVFileRecord fileRecord = new SQLServerBulkCSVFileRecord(csvFileName, true);
fileRecord.setEscapeColumnDelimitersCSV(true);

i.e. turning on escaping activates the handling of quotes, and conformance to RFC4180 on parsing the CSV data.

thanks to Andreas Radauer in the other answer (see here) for inspiring this answer.