The League should absolutely plan to take down Batman. They know for a fact that he controls an arsenal that allows him to render them worthless with trivial ease and HE'S the one member of the team with a history of mental instability. If I'm any one of the League, I'm coming up with my own Batman countermeasures, and I'm keeping my finger on the trigger every second that guy is in the same room with me.
Is Batman right to have anti-League countermeasures? Yes. The problem is that he is often the ONLY person on the face of the Earth who can access them. That presents two problems.
1.) If Batman is the only one who can access his countermeasures, then any Leaguer who DOES go bad is going to know "Take out Batman first. No Batman? No countermeasures to my nefarious plans."
2.) If DC ever had the guts to do a story where BATMAN is the one who goes bad (and given his mental instability, this should be a very real possibility,) then Batman goes bad and instantaneously has access to everything he'll ever need to crush the League in seconds, leaving him free to run unchecked.
One of the best examples I ever saw of a Batman countermeasure story was Injustice 1. Batman had a Kryptonite laser to handle Superman if he ever went bad. The catch? He couldn't access the laser by himself. He needed the cooperation of several other League members just to gain access to the weapon. Now, yeah, I think it was a bit excessive to require the cooperation of.....what? Five other very specific heroes? Something like that? If Superman ever went bad, there's no guarantee there will BE five specific League members left alive to help Batman gain access to the laser. But the mere idea that he decided that stopping Superman is not and should not be a decision he can make all on his own speaks of a much higher level of respect for his teammates and some actual forethought into preventing the possibility of the responsibility being abused.
I tend to have a problem with Batman having countermeasures for the League exclusively because those stories almost always paint Batman as the sole arbiter of whether or not the League needs to be stopped. He gets to sit in judgement of all the other heroes, and no one else's opinion or feelings matters. Add the very real prospect that Batman may succumb to his own paranoia and megalomania one day and turn those weapons against the League himself, and you've got a recipe for disaster.
The typical Batman countermeasure story shows a Batman who seems to have NOTHING but contempt for the rest of the League and he seems like he's just waiting for them to give him an excuse to beat them down because he's BATMAN!!! Giving the other members some say in how and when his countermeasures are deployed reduces the risk of them being deployed incorrectly and shows that Batman actually DOES respect his teammates. A fact that often seems to get lost in a lot of his stories.