Possible to ship logs to database with different name, but same structure?
It's been a while since I've used the GUI, however you can do it using TSQL with the command sp_add_log_shipping_secondary 

With this command you would specify the name of the secondary database and the primary, which would then restore the transaction logs to the relevant database name.

EXEC master.dbo.sp_add_log_shipping_secondary_database   
@secondary_database = N'ProdDatabaseNameReporting'   
,@primary_server = N'YourPrimaryDatabaseServer'   
,@primary_database = N'ProdDatabaseName'   
,@restore_delay = 0   
,@restore_mode = 1   
,@disconnect_users = 0   
,@restore_threshold = 45     
,@threshold_alert_enabled = 0   
,@history_retention_period = 1440 ;  
GO  


Going through the full TSQL implementation of Log Shipping using TSQL from https://msdn.microsoft.com/en-us/library/ms190640.aspx would be a good place to start for you.