I don’t think that’s possible. From what I’ve seen in Godot 4.7, the Asset Store tab is anonymous, so there would be no way for it to know you are the owner of the asset to show you even when unlisted.
You might have already considered this, but a simpler way would be using the Import button in the Asset Store tab to import your local zip file. Not exactly the same experience, but besides the download step, the installation would look the same as downloading directly from the store. For the asset store you’d have to create a zip anyway to push your code, so you might as well use your local file, self host it or even download the zip from your version control directly.