Initializing Base class and member using the same pointer without UB

project.get() is unsafe to me (and can just be project).

To fix your problem, I'd do this:

struct HasProject {
  std::uniqueptr project;
  [[nodiscard]] Project& makeProject(std::stringview path) {
    assert(!project);
    project= std::makeunique(path);
    return *project;
  };
};
class FakeResourceManager : private HasProject, public ResourceManager
{
public:
  FakeResourceManager()
    : ResourceManager(makeProject("/some/path"))
  {}
  virtual ~FakeResourceManager(); // defaulted in cpp
  void markResourceAsDirty(); // 'extension' of base functionality for testing
};

We can handle failures of project construction within makeProject (throwing or whatever), and don't have a method that takes a uniqueptr but fails catastrophically on an nullptr.