Simple. בבקשה means please. It is a word that people use. נא\אנא are relics of Olde Semitick. They can be found in poems, historical novels, ATMs and answering machines. "נא השאירו את הודעתכם", "אנא הקש את הקוד", etc. Your usage is correct, however, as נא precedes the request and בבקשה usually follows it (should be separated by a comma/פסיק, but it's often omitted).

שלום, נא, אנא זה לשון תחינה, כמו בבקשה (please) אבל בתחנונים. ו- זה כמו "and" באנגלית.