What's the best way to profile the GPU usage of a Direct3D11 game (to which we have the full C++ source code) and find out what might be causing dips in frame times or FPS? (including instances where shaders are contributing to this). Nvidia Nsight only does D3D12, not D3D11 so that's not an option. Is there any tool out there that can help with this?

RenderDoc is probably your best bet for DX11: https://renderdoc.org/docs/getting_started/features.html#d3d11​,​ although you won't necessarily have trace capability.