Hello,
Sorry, but we don’t have such functionality for moving the “Login/Register” button in the mobile sidebar, because this button comes from WooCommerce and we can’t influence it.
The feature when the button shows “Hello, Username” to the logged in user is available only for the header’s “Login/Register” button.
https://gyazo.com/06e793677a66f2479606cb2a308ea31f
You can enable it in WoodMart -> Header Builder -> edit the “Account” item and set the “Only text” option with enabled the “Show username” option.
https://monosnap.com/file/HugFI9ot1qycsyrjjEE0NqNPSVpFQy
Kind Regards