I was having the same problem. Everything was fine on Visual Studio and Unreal Editor but when packaging the header files was missing.
I was able to solve adding more detail to my include paths as @vr_marco mentioned above. In my case , I changed this:
#include "sqlite3.h"
to this
#include "../SQLLite3/sqlite3.h"
then I started to get an error about the DLL missing that I solved here Packaging Plugin with third party Dll - C++ Gameplay Programming - Unreal Engine Forums