Cohle wroteJust Manbat.
Lots of superheroes become more than what they are so much so that they are essentially a different hero, but you don't call them their original name followed by the new name.
Different property but The Hulk isnt Bruce Banner Hulk.
Or Kluh isnt Hulk Kluh. He is just Kluh.
Superman Prime isn't Superman Superman Prime.
Old man Logan isn't Wolverine Weapon X Logan Old Man Logan.
Angel got his metal wings and became Archangel, but we don't call him Angel Archangel.
The list goes on really.
This was just a silly joke in the show which people look too far into.
Yes but it does hot strictly work like that from how you have put it,
Superman Prime and Superman are the same.
Hulk and Bruce Banner are the same.
Archangel and Angel are the same.
And so on, but Batman and Man-Bat are two completely different characters.
If you're talking about Batman having Man bats powers and dressing up as a Man bat then thats not really different to Angel upgrading his wings and power set.
Its not really different to Logan getting his adamantium skeleton and becoming weapon X.
But if you're saying that Batman is just impersonating this other person called Man bat then lets look at it that way.
Think of it like Hush impersonating Bruce Wayne. If Hush is impersonating Bruce then do we call him Hush Bruce Wayne or just Hush (we as the reader know who is who.)
We'd just call him Hush.
In the Arkham games Alfred calls him Hush and Batman says 'Hush is impersonating Bruce Wayne', not whoever he's impersonating followed by Hush.
At a stretch I'd call it Batman Manbat, but at the end of the day it is just Batman or Manbat.
But if I had to pick one I'd go for Man-Bat-Man-Bat-Man because it rolls off the tongue better I think.