Uxtheme: draw combobox chevron without border

See more:

C++

Windows

UX

I'm trying to draw a custom control that should use the "combobox" theme class.

Using

m_hTheme = OpenThemeData(m_hWnd, _T("COMBOBOX"));
    auto stateBG = ...; // depends on window state
    DrawThemeBackground(m_hTheme, ps.hdc, CP_READONLY, stateBG, &clientRect, nullptr);

gives the correct background (read-only-look) without the chevron. But how do I add the chevron?

auto stateCV = ...; // depends on window state
    DrawThemeBackground(m_hTheme, ps.hdc, CP_DROPDOWNBUTTON, stateCV, &rect, nullptr);

draws the chevron correctly, but with its own border and the chevron centered within the `rect`. So if I use the full client rect, I get this:
combobox centered chevron[^]

If I use a smaller rect so that the chevron is positioned correctly, I get a separated dropdown:
combobox boxed chevron[^]

How do I get the "normal" look? - i.e like this:
How it should look[^]

Is there any documentation that does a better job than MSDN? It's as sparse as most newer documentation, e.g. just listing "Parts and States", without describing their purpose (which is not always obvious), and whether it's `DrawThemeBackground` or ~`Edge`for a particular item.

Bonus Question: Do I still use the good old `DrawFocusRect`for the focus rect?

What I have tried:

(see above. I've tried a lot. like really a lot. 30 chatracters you say?)