I sympathize with your member's heartfelt desire to know this, I'm sure it's truly meaningful during their faith crisis. However, from the father's perspective, I think it would be incorrect for the Church to share this information. Imagine somebody who worked to get their membership restored, and to get to that state where the Lord says about their sins that He will "remember them no more" -- to have others be able to access his past membership status without his permission. He of course can share with whomever he wants while alive, and if he shared with nobody before passing, then I'm not sure how proper it would be to share after death.
Of course the Church at some level does "remember", since while alive a member's Bishop or Stake President may have a need for this info if subsequent membership councils are needed. But this is only provided by HQ for approved situations, and unlikely for a family member who wants "to know".