As far as I can see, DirectX11 (and probably 12, too), perfectly support RGB565 textures via DXGI_FORMAT_B5G6R5_UNORM.
Adding
case SDL_PIXELFORMAT_RGB565:
return DXGI_FORMAT_B5G6R5_UNORM;
to SDLPixelFormatToDXGITextureFormat and skipping over the IsSupportedFormat call in SDL_CreateTextureWithProperties was sufficient to get a working texture that gave me a ~3x perf boost due to offloading color conversion to the GPU.