Tossing it into Sag A* might work, but it's both rather tricky to pull off, and renders the phylactery inaccessible if it survives... Best to stick to methods known to work.
^This, really. We can spitball all the ideas we can come up with and some of them
might work, but that is no guarantee. Especially when it involves a Major Artifact made by a God with a known and specific domain, the capital 'G' is very intentional here. In short, the artifact would need something that is the complete antithesis of the artifact, and of more power at a fundamental level than its creator, for it to be destroyed. And due to Fakazure, the creator of it, being a Major God, the requirements get even
more specific.
In this case, it means that the phylactery would need to be on the same world it was created and via a very specific means in order to ensure its destruction. Are there other ways that might work? Yes, but it is not guaranteed that they will.
Think of how an artifact created and imbued with the eseence of Yahweh, the Abrahamic
God would need to be destroyed, and it was created on what is Humanity's First World. Given the multiversal nature of the setting, and how Humanity has this annoying tendency to wander and get to the damndest places, finding
that world would be an extremely tall order unless you already knew exactly where to look in the first place. Fortunately for our Heroes, they have someone who knows where to go, even though she can't get there herself. She knows the way and can get everyone else on the right path.
Say what you want about Falazure, but he
is clever here. With Io having barred the First World to dragons, and that means
all dragons, it is protected from destruction at the hands of the heaviest hitters in the story for the time being. If Io ever breaks free of his captivity, that can change, but he is the only one who can rescind that restriction, and he might not. Because things like that tend to be all or nothing affairs, and rescinding it may mean Falazure now has access to the First World again.
I think you can see how that would be suboptimal, to put it mildly.
Destroying the phylactery is simple, we know the requirements. But just because the requirements are simple does not mean that meeting those requirements will be easy.