That's the key point, "Ultimately, it is up to management to decide where SQL resources like DBA labour hours are spent."

This is a resource assignment issue. Just calmly explain to management (whomever that may be in your world) how your time is pulled away from other things to work on something like this. And explain how you could avoid that managing the server up front. Let management sign off on an arrangement. Then your hands will be clean. You may not avoid all frustration with future incidents, but at least you will have done your part in raising this issue with management.