It used to be polite in formal contexts, instead of saying "sorry". In some remote corners of England it might still be thus but for the majority of us its only use is to suggest a criticism of the other person. It implies that the other person has been badly behaved and can be said with outrage or fun in your voice, depending on who you are speaking to.
It sounds very formal, and I'm not sure it conveys what you want to say. "I beg your pardon" can be used apologetically when you bump into someone or didn't understand what they said, but it's not a way to apologize for rude behavior, poor service, burned food, or a customer's bad experience.
Florentia has been firmer than me is saying Do Not Use This. I would not argue with her, my recent opinion was only tentative and four years ago I was pretty sure it had gone our of use! So yes, avoid it!