Is Batman a compound noun since it is made up of two nouns? Or being a name it remains only a proper noun?
I would argue that proper nouns cannot, by definition, be thought of as compound nouns, even when they look like they are.
If we look at a compound noun like "doorknob", we see that we can break it up into two separate nouns that each contribute meaning to the final word. It is a knob for a door.
You wouldn't break apart someone's name to derive meaning, though. If my name is "River Bush", this would not imply that I am a flowing body of water or a type of plant. It's just my name. You can't break it up to derive meaning, as if it was a compound noun, even though it looks like it could be. Similarly, if my last name was "Toothpaste", I would still suggest that cannot be considered to be a compound noun. It shares the same letters with a commonly used compound noun but it's my name. You cannot break apart my name to derive meaning about me and therefore my name cannot be a compound noun.
So I would say the same for Batman, even though he actually is a man who has a proclivity towards bats. His name is formed from a compound noun but once it becomes a name, it is a proper noun and the guidelines for compound nouns no longer apply (you cannot decide to spell his name as Bat-man or Bat Man or even necessarily assume he has anything to do with bats, because it is just a name).
A compound noun is the sum of its two parts. Just because you can divide a word into two other words doesn't make it compound.
Batman as a name is still a compound noun, just as it is when it is a common noun.