If your application is written in a Windows 10 conformant way, it should pick up it by itself. You should not need to change any code.
"Use manifest" applies a generic "standard" manifest and you can change very few options of it (mainly DPI awareness, etc.).
To apply a more carefully drafted manifest you should uncheck that option and add the manifest file as a resource to your project ("Project Options->Resource"). You might also want to add it as a project's file in the Project Inspector so the tools know it's there.
Can't offer much more help; I left Windows behind, other than for testing and very few other things, quite some time ago. :-\
Did you try setwindowtheme with the handle to your control?
I don't know if the most recent W10 update changed this, but W10 dark theme used to be for MS apps only.
The introduction of this theming library also affected the common controls library, COMCTL32.DLL. For compatibility reasons, Windows XP (and all newer Windows versions) are equipped with two different versions of COMCTL32.DLL.
At the standard path C:\Windows\System32\, the old COMCTL32.DLL version 5 can be found. This library version contains the implementation of standard controls like list view, combo-box, etc., which are unaware of UXTHEME.DLL existence, hence applications linked with this library generally are not themed.
The newer version 6 of COMCTL32.DLL resides under C:\Windows\WinSxS\, available as a side-by-side assembly. Only applications explicitly manifesting their compatibility with the library version 6 use it. The other applications simply continue to use the version 5.
Also note, that historically some controls used to be implemented in USER32.DLL (window classes like, for example, BUTTON or EDIT), while others (known as common controls) resided in COMCTL32.DLL. This has changed with the introduction of COMCTL32.DLL version 6, and now all theming-aware controls live there.
Gotcha: The old (unthemed) implementation of standard controls is still available in USER32.DLL. If you instruct the linker to link your application with COMCTL32.DLL, it may silently omit it if the application never calls any function from it. Hence I recommend you to call InitCommonControls() when initializing the application. Otherwise, the app may be just unthemed instead of not working, using the old controls, masking perfectly the root cause of the problem.
As marcov wrote, the problem is that there is no official API yet to enable dark mode for native WinAPI controls, the applications provided by Microsoft (e.g. Windows Explorer) simply use inofficial, undocumented ones. This (https://github.com/ysc3839/win32-darkmode) C example shows what needs to be done, though I don't know whether applying these techniques to the LCL would already fix the whole of the LCL or if other things would need to be adjusted as well.