If Batman met a clone of himself or a Batman from an alternate dimension, would we say:
There are two Batmen
There are two Batmans
You'd assume Batmen but Batman is his name, not his title.
Have we ever seen any examples in canon?
There have in fact been multiple occasions in the comics where Batmen was used as the plural form of Batman, one such instance being the "Batmen of All Nations".
As a footnote, aside from instances in which the Bruce Waynes of multiple dimensions have encountered each other, there have been several individuals to take up the mantle of Batman over the years in various continuities, such as Jean Paul Valley, Dick Grayson, and Terry McGinnis.
Grammatically, while Batman is used like a proper noun, its derivation is a descriptive conjunction of two common nouns, formed in such a way that the noun "bat" serves like an adjective as a descriptive modifier to the common noun "man," because, well... we can skip that explanation right? Oh good.
So it is in fact totally fine to use the common plural of "man" (e.g., "men") and conjoin that once again with "bat" to arrive at "Batmen." We even have other answers here showing that this is a canonically accepted form.
But one consideration that no one has met is that the idea of many Batmen is slightly ludicrous. And "Batman" does indeed serve as a proper noun contextually. Thus, "Batmans" may actually be the right wording in particular contexts when wanting to call attention to the comical nature it may invoke, given how—technically correct or not—clumsily wrong it sounds.
"There is a bevy of Batmans headed our way! What is this I don't even?"