Uh, I accidentally deleted my Conquest file, but I still have the mark that said I completed the game on that mode on my card. The castle and the card information (personality, home, etc.) is overwritten, but the number of visits and battles you attained is kept.
So to answer your question, the medal does pass on to your entire game. Example: I completed Revelations on Normal to pass on skills and I deleted that copied file. On my card, it shows I completed Revelations on Normal.
I currently have an unfinished Lunatic Conquest file and unfinished Lunatic Revelations file only. No Hard Conquest or Normal Revelations files, but I still have the medals for completing them.