Thanks for the detailed rundown, Lucas! this is exactly the kind of report I was hoping for. Glad it launched once you found a Proton version it liked.
The drag-and-drop freeze looks like a limitation of Wine itself rather than something in the app. Dragging files from a native Linux file manager into a Windows program under Wine has been a known issue for years. On Windows it works normally. Under Wine, the Load audio button (and the Batch folder button for whole folders) is the reliable way to go, which you already found.
Once you export, everything the app writes is just ordinary files, so the WAVs and the Godot script will behave the same as they would from a Windows machine. I'd love to hear how the Godot side goes.
With your permission I'll include what you found into a short "running under Wine" note so other Linux folks know what to expect. It works under recent Proton, use the button instead of drag and drop.