Using/storing derived member in derived class with base class that stores base member

A situation I often come up against is having a set of classes, Base and Derived, where the Base class has ownership of a base-class member BaseMember, and the Derived class has a reference or pointer to the same object, but as a DerivedMember.

For example, a UI panel class that contains a specific instance of a certain type of control with some special-control functions, inheriting from a general class that contains a general control and has general-control functions.

First, say that BaseMember is inherited by DerivedMemeber.

Without using smart pointers, I might do something like this:

class Base
{
protected:
    // receive ownership but only because we say so,
    // someone else can still try to delete as it's "just a pointer"
    Base(BaseMember* _bmember):
        bmember(_bmember)
    {}

public:
    virtual ~Base()
    {
        // perform an owner's duty
        delete bmember;
    }

    // functions that might be based on BaseMember + other base state
    void SetMemberId(....) 
    {
        bmember->SetId(baz);
    }

private:
    int baz;
    BaseMember* bmember; //owned, but not smartly
}

class Derived: public Base
{
public:
    Derived(DerivedMember* _dmember):
        Base(_dmember),
        dmember(_dmember)
    {}

    // functions that only make sense for Derived + Derived/Base state
    void SetDerivedFrobulation()
    {
        // only a DerivedMember has frobulation, so only
        // Derived allows users to access it
        dmember->setFrobulation(foo);
    }

private:
    int foo; // some state
    DerivedMember* dmember; // no ownership here
}

With smart pointers (C++11 and up, specifically, I don't really care about older C++ in this case), I am tempted to do something like this and never let the Base/DerivedMember object out into dumb-pointer-land where it could leak if there was an exception somewhere inconvenient.

class Base
{
protected:
    // receive ownership
    Base(std::unique_ptr<BaseMember> _member):
        member(std::move(_member))
    {}

    virtual ~Base()
    {}

public:
    // public access functions here as before

private:
    std::unique_ptr<BaseMember> member;
}

class Derived: public Base
{
public:
    // pass the ownership down by unique_ptr
    Derived(std::unique_ptr<DerivedMember> _dmember):
        Base(std::move(_dmember)),
        dmember(_dmember.get()) // _dmember is moved! SEGFAULT if access dmember later!
    {}

    // public access functions here as before

private:
    // handy handle to the derived class so we don't need to downcast the base (or even access it!)
    DerivedClass* dmember
}

As I noted there, you can't "steal a peek" at the DerivedMember class as it comes in to the Derived constructor, because the unique_ptr is moved away before Derived gets a look in.

I can see a solution in providing a protected access to the BaseMember and static_casting back to DerivedMember in the Derived constructor (i.e. after the Base constructor is done), but this seems an ugly way to get access back to a variable we let slip though our fingers!

Another way could be each inheritor of Base owns the pointer, and base just gets a dumb pointer. In this case, the Base destructor doesn't get access to the member, as it's already gone. Also it would duplicate the ownership logic needlessly.

I think either:

This is symptomatic of an anti-pattern and the design of the whole Base/Derived/BaseMember/DerivedMember system is not good practice.
I'm missing a trick and there is a clean way to do this without fumbling a smart pointer and making a leak possible or adding functions and exposing interfaces or casting too much.

Is this a good pattern for re-use, or should I look elsewhere?

Expanding on the use case (EDIT)

In a core library, I have a class DataInterpreter which shows "some interpretation" of data - could be a string, an image, etc. This is then inherited by, amongst others, TextInterpreter which presents a string.

I then have a DataDisplayPanel class which represents a piece of UI for displaying in an abstract sense. Exactly what is in this panel will depend on the interpreter used: a TextInterpreter should get a text entry field and say a button to set some text display option, and that is handled in TextDisplayPanel, which has "special" knowledge of the text aspect of the interpreter.

There is then a DataAggregatePanel which combines a number of DataDisplayPanels and provides some global settings that affect all displays (via virtual functions), and manages the panels in a std::vector<std::unique_ptr<DataDisplayPanel> >. This aggregate class doesn't deal with any of the derived classes at all, any functions would be polymorphic and defined in the base.

In the application (which depends on the core library), these classes are extended (by inheritance or composition, whichever makes more sense). For example, if the application is a WX GUI, I might have wxDataAggregatePanel which contains wxTextDisplayPanel (and others), all of which are wxPanels. In this case, wxTextDisplayPanel might own a wxTextEntry and either own or inherit TextInterpreter and use its knowledge of the TextInterpreter's specific methods to fill the text box with a string.

Expanding on the use case (EDIT)

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