How to manage the scope of db context in .NET Core when SaveChanges() fails

My understanding is that the default lifetime of the DbContext via DI is Scoped and that it's recommended to let DI and the framework manage the lifetime instead of using using statements.

This is not accurate, DI is a method for instantiating objects, but you can still control the scope of the objects... example using Ninject (more info):

// this would create a new MyType instance on every request
kernel.Bind<IMyType>().To<MyType>().InTransientScope();

// this would share MyType instance within the same Thread
kernel.Bind<IMyType>().To<MyType>().InThreadScope();

// this would share MyType instance within the same Web Request
kernel.Bind<IMyType>().To<MyType>().InRequestScope();

Regarding the scope of DbContext, see Microsoft's recommendation:

A DataContext is lightweight and is not expensive to create. A typical LINQ to SQL application creates DataContext instances at method scope or as a member of short-lived classes that represent a logical set of related database operations.

So it is generally a good idea to share the same instance of DbContext within the same method... the exception here is with logging. It is not a good idea to mix your Logger's DbContext with the operations that it is logging. Logging should be independent of these operations... you many choose to log your message in a text file, email, DB... if you want to use DB for logging, you can consider using a different DB for your log message. Also Consider using logging frameworks such as Serilog, NLog, etc