Are you getting denials from Medicare? They should be telling you what is going wrong.
You are getting confused about the Part A and Part B. You only need to know that the patient does have Part B coverage. You use the only Medicare ID they have no matter the letter or letters on the end of the number.
If the denial states name and ID do not match, make sure the name is entered on the claim exactly as it appears on the ID card. If a middle initial is on the card, use it. Spelling has to be exact.