As it turns out, this is in-built functionality.
Go to your animation tree and right-click your playback node (the state machine root) and click "save"
In your new animation tree (blend tree in my case), right-click anywhere and click "load..."
I tried this when I first met this problem. Turns out I just saved it wrong. Hahaha.
Godot is so convenient. A little bit of trusting in that fact can pay off.