I agree with @Steven's approach, but that doesn't answer your more general question of how to resolve per request.
I would recommend you change to using the UnityHierarchicalDependencyResolver and then anything you register with HierarchicalLifetimeManager will be resolved per request.
Change this...
GlobalConfiguration.Configuration.DependencyResolver =
new Unity.WebApi.UnityDependencyResolver(container);
to this...
GlobalConfiguration.Configuration.DependencyResolver =
new Unity.WebApi.UnityHierarchicalDependencyResolver(container);
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