Pronoun problems!
I agree with you that the answer should be "He begged me not to tell you about it."
1B: He begged me not to tell it to you.
This is wrong because the verb "tell" uses the recipient as the object, unlike how "say" uses the message as the object. You tell someone (about something), but you say words (to someone).
2B: He begged me not to tell that to you.
This is a bit closer, but by phrasing it using "tell that to you" it implies that the speaker is telling lots of other things to you, and was only being asked to omit this particular one.