It feels like it'd make all the sense in the world: the moment to arrange your escape plan is when you're free and with all your resources at hand, and the efficiencies of division of labour make giving money to experts the most effective means for that — there could be any number of specialist service providers vying for the contracts, each of them with pre-bribed (or blackmaileable) prison personnel, already hacked and breached security systems, etc. (Would there be any honest, or not absolutely terrified and malleable, employee in Arkham? At all?) Compared with the standard, ad hoc prison break, it should go well much more often.
The League of Liberators? The Early Release System?
The only problem I see with this idea, at least during the (in-universe) contemporary DCU, is that such an stable organization — a group that does and keeps doing the same thing for long, with systems for receiving and paying money, and so on, even with the benefits of the best encryption in the world and a decades- (or centuries-?) long reputation for unimpeachable trustworthiness — wouldn't stay out of Bruce and Babs' radars for very long, and once they find about them, the strategically obvious thing to do would be to dedicate their entire resources to bringing them down.
It's one thing to decide you don't have the time to bring down every criminal organization you know of, so you hurt them as much as you can whenever they are near the top of your list, next to the world crises, criminal masterminds, innocents in immediate danger, former allies trying to kill you, ethically dodgy and overly ambitious things you did for arguably good reasons coming back to bite you in the ass, etc. But wiping down these guys for good would make everything else vigilantes (and actually law enforcement forces) do quite a bit more effective, if nothing else extending the time the really dangerous guys stay in prison, which is probably one of the most time- and cost-effective things you can do to reduce the overall levels of serious crime.