Transfer database backup from Production to Staging during business hours
Usually copying your latest database backup (from the production backup location) to an area where you can restore it over the top of your staging area should have no effect on your production database.  The only time where this would affect it is if you're storing the backups on the same drive(s) that is hosting the live database, but hopefully you're not.  In this instance, the file copy operation will just generate extra, and competing, I/O with your normal database operations.

Your question doesn't provide enough context around how this sync process works, but a 15GB file copy isn't usually large enough (in my experience) to crash a server or cause any severe performance degradation.