You could make internals visible to Moq by adding InternalsVisibleToAttribute in your project’s assembly.cs, like this:
[assembly: InternalsVisibleTo("DynamicProxyGenAssembly2")]
Why "DynamicProxyGenAssembly2" and not "Moq"? It’s the name of dynamic assembly created to contain dynamically generated proxy types (all of this is handled by yet another library, Castle’s DynamicProxy) which is used by Moq. So you expose types to dynamic proxy assembly, not to Moq itself.
But, what’s the point of mocking class if there’s no overridable member? You won’t mock anything and all calls will use actual implementation. Your second solution,
I guess I could create an interface (say “IClassB”) that ClassB implements, inject that into ClassA, and mock the interface instead.
is what I would normally do. Its purpose is much more than “to support unit test mocking” – it helps you build losely coupled components, which is always something worth aiming for.