A customer is using our dll which is creating a child process which uses an open source library, which ultimately fails because of a call to LoadLibraryExW(), the last error
The best way is to use loader snaps. Basically you use gflags.exe (which is included with windbg) to enable loader snaps; then, run the process with the debugger attached. Loader snaps will enable the loader to print out dbg messages of the process and it will print the failures.
gflags.exe -i yourcode.exe +sls
windbg yourcode.exe
Your dependencies might be present on the system but they could be in a folder that is not part of the search order during LoadLibraryExW(). A SetDllDirectory() or AddDllDirectory() call would ensure that the folder containing the dependencies is searched during the LoadLibraryExW() call
Use the Profiling option of the Dependency Walker on your application. Possibly the library is trying to resolve some APIs dynamically (using LoadLibrary/GetProcAddress) and this won't show up in the static dependencies.