Hey,
There have been some issues reported in previous threads around upgrading from 7.11.13->7.11.15:
After the upgrade fails, do you get any particularly notable errors in the upgradeWizard.log file?
In terms of moving a db to another instance, I think you should be fine.
Although, it would still be worth taking backups of both the instance and the database beforehand, just in case.
One thing to note is that if you link the CRM to the database before running through the CRM install process, it may override the database and generate a clean one in the same name.
(In essence, wiping the database)
So I would suggest running through the install for the 7.11.15 version as standard
Then, once installed fully, replace the database values in the config.php file to point towards your target database
Or, depending on how much data you have, you could also export the data from the 7.11.13’s database, and import it into the 7.11.15’s database
Let us know if you come across any issues!