MembershipProvider with Entity Framework and IoC
The preferred method of doing this is to use a factory dependency that creates the DbContext for you. So instead of the DbContext as the dependency you have a ContextFactory as the dependency and its consumers can get and manage a local DbContext by calling contextFactory.CreateContext();

If you insist on having your context injected directly, I believe it would be much better to have your MembershipProvider be registered per web request (even though you don't have control over creating it, you still have to register it right?). It shouldn't be a bottleneck. From a design and testing perspective this is much preferred to having a hard dependency on a service locator. How will you test the class by itself? Would you be incorporating your service locator into your tests?