When it comes to Gotham City, Batman is very protective. Yes, he's the protector of Gotham but that also means the city is his responsibility and no one else's in his mind (unless they're bat-affiliated such as past or present Robins, Batgirl or any other Bat-Family member). Other than that, Batman sees no reason why any other hero should give him a hand or help him out, least of all the Justice League.
Batman is a fairly isolated and lone-wolf hero, despite being a founder member of the League. When it comes to saving the entire world, the Justice League are of course welcome allies to have. However, where Gotham is considered, Batman seems to have made it clear that he doesn't need their help at all... But, there was that one time though.
Back in 2016 at the very beginning of DC's Rebirth initiative, Batman #5 featured a rare (and perhaps singular) moment where Batman needed help in Gotham. Essentially, two new heroes came to serve the city, Gotham and his sister Gotham Girl. While each had incredible powers and gifts, they were both young and inexperienced. Batman determines to train them and to help them be better heroes. However, due to the manipulations from Hugo Strange and Psycho Pirate, Gotham and his sister become compromised. While Gotham Girl becomes paralyzed by fear, Gotham himself goes insane and is thrown into a massively destructive rage, one that Batman is not very well-equipped to stop on his own, seeing as how Gotham and Gotham Girl have powers on par with Superman. Speaking of Superman...
The level of destruction and massive endangerment to citizens is what causes Batman to make the call to the Justice League for aid. However, this isn't before his butler Alfred had to put on the cape and cowl to stall Gotham, and Batman had to intentionally crash his jet into Gotham (to little effect). Finally, Batman tells Alfred to make the call. Alfred is stunned, asking Batman if he's sure, as they've never called the League for help before. Batman confirms that he's sure, and sure enough, the League arrives in no time at all.
Of course, the League is happy to help, though for Batman it was probably a hard call to make given his protectiveness over his city. Even so, Gotham gives Batman and the League a run for their money, almost taking them all down if not for the intervention of his own sister, who recognized the need for him to be stopped. It's moments like this where one would think that Batman would open up more and ask for help. However, this specific moment still remains one of the few times the entire League has come to help Batman deal with a problem. Will Batman ever get over his stubbornness? Probably not. His cold, dark, and brooding isolationism is pretty much one of the core things that make him Batman in the first place.