A Scheduled Task requires you to specify an account for running the task.
Each action the task performs will be executed in the context of this account, even commands like net use.
Example for running a Script with Task Scheduler: here
net use even allows you to connect a share with a different account.
copying itself is in my opinion best don by the good old robocopy, especially for larger transfers.
Using Powershell in a user context, which has the correct permissions, you would be able to use Copy-Item with UNC paths and therefore it wouldn't require you to first connect a drive. (which would by the way also be possible with robocopy)
There are al lot of finished solutions to this topic which you can easily find online.