I think what is happening is that there is a slight bug when the multisite network is in subdirectory mode. It looks like the permalinks are not updating to the new URL.
You should be able to work around this by going into Settings > Permalinks and clicking Save Changes (without changing your settings). This will force WordPress to update the permalinks for that site and should fix it and allow you to use the cloner until we can get a patch for it completed.
Step 1: Clone as normal.
Step 2: Settings > Permalinks > click Save Changes.
Sorry for the additional step – it will be unnecessary once we fix this bug, but hopefully it will fix your issue for now.