It would probably show the same name twice. Because unless his name is registered as Bartemius Crouch Jr., there are 2 Bartemius Crouch's. And yes the map can distinguish between the 2 of them probably. But its just a map it tracks people and puts there name down. It doesn't know all of their personal business. But logically if two people have 1 name how does the map know to add little hints(like add Jr., Sr., or Mr. etc.) as to who the person is.
This might help.
One of the Map's possible flaws lies in the fact that it cannot
differentiate people who bear similar names (the shared names aren't
given a Junior or Senior at the end)
Harry was saved by Barty Crouch Jr., disguised as Professor Alastor
"Mad-Eye" Moody, who pretended it was his. Due to the map's flaw, it
could not differentiate between the Junior and Senior Crouch. Crouch
later asked to borrow the map from Harry, but after Crouch was
unmasked, he reclaimed it.
"The magic used in the map's creation is advanced and impressive; it
includes the Homonculous Charm, enabling the possessor of the map to
track the movements of every person in the castle, and it was also
enchanted to forever repel (as insultingly as possible) the curiosity
of their nemesis, Severus Snape." —Description of the magic involved
in the map's creation