Didn't Arrow already drop a Barbara Gordon reference? Something about the hacker name "Oracle" already being taken?
I think that was just an in-joke by the writers, a wink to the audience to lampshade the fact that they couldn't get clearance from DC to use Oracle as Felicity's code name. It doesn't mean they literally intended for Barbara Gordon/Oracle to exist on Earth-1. After all, that would be a bit redundant, since Felicity already is the equivalent of Oracle, even to the extent of going through a very compressed version of Barbara's paralysis and recovery.
That plus I there was an obvious allusion to Batman on Supergirl not that long ago.
In that case, it seems more clear that it was explicitly intended to indicate Batman's existence, especially since they've name-dropped Gotham a couple of times.
Of course with a multiverse and time travel a staple of this franchise they can pretty much do anything they want including a classic 1940's era version of Batman, guns and all.
Batman made some very infrequent use of guns in his first year, from 1939 to early 1940, but that was cracked down on before the character was even a year old. So despite the revisionist history advanced by some modern fans, it isn't accurate to say that 1940s Batman was characterized by gun use. That was an artifact of the character's rough beginnings when he was basically just a stock pulp crimefighter and hadn't yet developed his distinct identity. And even in that first year or so, although he killed a lot of villains through various means, intentional and otherwise, he only rarely used guns to do it, and usually against nonhuman foes like vampires. More often he did things like kicking a fight opponent off the edge of a roof, snapping a shooter's neck with a kick, swapping clothes with an unconscious guard so that the guard would get mistaken for him and shot dead, and in one rather ghastly example, pushing a giant statue down onto a whole bunch of caricatured Asian villains. (Or was that Superman? I forget.) But guns were never a regular part of his repertoire even at his pulpiest.
For most of the 1940s, Batman was a fairly upbeat, heroic crimefighter partnered with Robin, the Boy Wonder (The Sensational Character Find of 1940!), engaging in witty banter with his boy sidekick as they fought evil with their fists, brains, vehicles, and gadgets. This version of Batman and Robin started appearing on the Superman radio show in 1945, although without much in the way of gadgets.
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