Tempt means "entice" or "encourage" you to eat or do something you are reluctant to do. Persuade is similar but means to encourage you by reasoning and argument, so that is where there can be interchangeability, in the take a bite context. To tempt someone to have a bite, you might use words as well as encouragement to describe why they should take a bite enticing/encouraging by actions and by persuading at the same time with words.

It's usually used when you want someone else to do something, and they are possibly reluctant. A waitress in a restaurant: "Can I tempt you with dessert?" (I'd like to sell you a dessert. We have good ones) One neighbor to another: "Can I tempt you with a cup of coffee?" (Would you like to come to my house for coffee? I'd like to visit with you and am offering you coffee to encourage you to agree)

Can I persuade/convince you?

Thank you!

Hey! It can mean a couple of different things. So there saying "can I tempt you with <item>" usually refers to food, particularly something nice or something very bad (so the saying is used in a joking manner).