I understand why Batman doesn't kill the Joker. But the fact remains that Arkham just doesn't seem to be capable of holding him. So why not have him sent to an interdimensional prison designed to hold Kryptonians?
I'm wondering if Batman has ever asked Superman to put Joker into the Phantom Zone? If so then why didn't it work? If not then why not? It would seem to be a perfect solution to taking Joker off of the table without having to kill him.
I'm not aware of any time he has, though I'll have to search further to check; but I'd be surprised if he ever did. (note: per a comment he did in the Lego Batman movie, but this version of Batman isn't exactly in line with the "main" comics one)
Sending the Joker into the Phantom Zone doesn't fall in line with Batman's way of thinking; there are several reasons he wouldn't want to. In no particular order...
Batman deals with his own problems. Asking Superman for help once in a while is a thing, entrusting him with putting an end to his feud with his nemesis is another.
Batman is a control freak. Everything that happens, he must have a handle over it. But he has no control over the Phantom Zone: it's Kryptonian-made, so it's less reliable than something Batman would have built himself, and while he can walk into Arkham pretty easily, it's not the same with the Phantom Zone. If something is happening with the Joker, he has to hop into a Batvehicle, reach the Fortress of Solitude, wait for Superman to open the thing, and act. Arkham is a short Batmobile drive away.
It sends the wrong message. When Batman puts a villain in Arkham, Gotham citizens know it and can relate to it: "we know that guy is behind bars now", "it could be me if I screw up". Arkham is a massive building which reminds you of this every time you walk by it. The Phantom Zone is something most Gothamites have never heard of, and probably would have too many other concerns to understand and care about. It's too far away from "normal life" to be something folks will consider when living their life: "Kryptonian prison? This is only for the biggest guns – me, I just robbed a bank and mugged a couple citizens".
Batman has no authority to do so. Theoretically, all Batman does is drop criminals at the gate of the police station or Arkham; the law process goes on without him after that. Being Batman, he monitors everything, but the point is that the people take care of the rest. The police takes convicts from cells to the court, the court decides if it's the chair or a stronger cell, etc. Batman drops the Joker into somewhere with Gotham police's jurisdiction; the Phantom Zone is not part of it. Batman wants to make it look like the thing is handled by society; not only the GCPD having to call Superman to bring the Joker out is contrary to that, it adds yet another layer of logistics for them to handle. Now sometimes the police will be very glad to let the Justice League deal with the most supernatural freaks, but the Joker is "only" human – if humans can't handle humans, then who can?
Out-of-universe it would make for a pretty poor story, too. Superman's prison becoming the go-to "in prison but can't escape" plot device for the whole DC universe to turn to would be lame.
