Hi Scott,
did you try to use "Pending Clear"?
If the problem is solved, the Alert can go to pending clear - being in that status it does not impact any Services anymore (it is exempt) and can be moved to another Queue (to not disturb Queues Operators look at) or be hidden.
You can create an Escalation Policy to automatically close these Alerts after x minutes.
As long as the Alert is in Pending Clear, no ticket will be created again, because the system only executes an Escalation Policy once for every Alert.
But you can assign new policies to the special Queue for "Pending Clear" to update ticket, .....
We can discuss further details if you want to have a go at this concept.
MichaelBoehm