by compatibility mode with X11, are you referring to XWayland?
Yes
And is that with the source application running under XWayland, the destination application, or both?
The problem arises only when running the application through XWayland, whereas on pure Wayland there are no issues
Is that red and white circled-X icon in your screenshot what the cursor looks like during that process?
And am I reading it correctly that the actual drag-and-drop can still occur, but the icon just visually implies that it couldn’t?
Yes both