This seems like a bug, or at least a missing feature.
Here's what's happening from a technical point of view. In "Append" mode (see Sidebar > BlenderKit > Import settings > Import method > Append), the add-on searches all objects in the active scene and checks for an "asset_data" object property with a matching "assetBaseId" component. If it finds one, it reuses that object instead of trying to append the original. If you have multiple copies of the asset that you've independently edited, it'll just copy the first one it happens to find. The master copy that you see when you right click and choose Selected Model > Open Directory should be the original, unedited version, but that doesn't matter because the add-on doesn't even access it if it finds another matching object already in your scene.
So, before doing anything else, check that you can successfully append the original model into a fresh Blend file. This will confirm that the "master copy" (the one that you see under Selected Model > Open Directory) is the original, unedited version.
Then, check the file that's giving your problems. Somewhere in your scene, you must still have a copy of that object, even if it's not currently visible. In the outliner, switch the view from its default of "View Layer" to "Scenes" and search under your scene's Objects for an extra copy that's sitting around, and make sure it's deleted (e.g. right-click > Delete from the outliner):

Once all copies in the scene have been deleted, you should be able to drag the asset back in and get the original
Update: This won't fix your current problem, but in the future if you want to re-import an original copy of the asset without having to delete all your edited copies, you can break the connection between objects and the BlenderKit asset by going to the object's Properties > Object > Custom Properties setting, and deleting the "asset_data" property by clicking the "X":

You have to delete this property on all existing copies in the scene. Then, when you re-import the asset, the add-on won't be able to match it up with any of the existing copies, and it'll append a fresh version.