I like to think this is one of those things Bruce had been trying to do, legitimately of course, for years. But somewhere between Gotham's corruption, well timed breakouts, and no other prison really wanting to deal with the Joker anyway; every prior attempt to transfer him fell through.
By swapping paperwork, Danny quite accidentally managed to cut through all those problems. The transfer was already happening before anyone knew anything about it. Therefore, it didn't need some corrupt officials to sign off on it; the receiving prison didn't have to be strong armed into taking him; and probably most crucially, Joker had no time to breakout between learning about his transfer and said transfer happening.
I also think it'd be good if Danny didn't even send him somewhere super high security (ie Belle Reve). Those transfer are probably rarer and come with too much fanfare. Instead, Danny sent him to some random prison in, like, Wisconsin* or something. Which, on top of being narratively funnier, actually works out better because it's so far removed from Joker's normal network of goons and corruption. He either has to wait for his normal helpers to get there, or try to build all that influence back from the ground up while being stifled by being under such close watch and having to fight his way to the top of whatever established hierarchy there already is in the new place (again, alone, without his own established network bolstering him).
Either way it's gonna take time. Months and years even. And, once they know about it, time is very useful to the Bats for cleaning house before Joker inevitably ends up back in Gotham. The more of the Joker's network they can root out and clean up, the harder it'll be for him to breakout again later.
Plus, the longer this random prison can hold him, it helps build very strong evidence for Bruce's legitimate attempts to get Joker transferred. No more other institutions arguing they don't have the means to secure someone like the Joker. Security isn't the issue; Exhibit A, this prison did it for years no issue.
*Danny, leaving a voicemail at 3am during his stressed out sleep deprived brilliance: Hey Frootloop! You've got plenty of influence in Wisconsin. Do something to help keep Joker in his new home and I'll go to 3 Packer's games with you every year he's there. Kay thanks, bye! Whoever invented statistics is evil. Evil, evil, bad box of Raisin Bran evil. Where'd my calculator escape to? *Several more seconds of rustling and muttering before Danny actually remembers to hang up.*
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