Copy database wizard-generated job blows up log file; alternative?
The package created by the copy wizard can be modified to bring a little better performance possibly but your log growth is expected. Every bit of data changed for the table and indexes is being modified, that has to be logged no matter what recovery model you are in.

I would go with the restore from backup as this will be a bit cleaner. You can pull the last backup file name from the msdb backup tables. There are a ton of examples of this online, would check MSSQLTips.com for a start.

You would simply pull the last backup, verify it exist, robocopy it over (unless it exist on network share), then restore it.