Error in Arduino Library file
The core of the problem is that the code is being compiled as C and not C++.

The error says 'new[]' is undefined at the link stage.  This is because the linker is not pulling in the C++ standard libraries.  The built-in function new[] it doesn't exist in C. 

The filename FPS_GT511C3.c.cpp is the elephant in the room.  You could consider renaming it (and others) to FPS_GT511C3.cpp.  Although the plain Arduino IDE does not really have a problem with building them.

Another option might be to modify the new[] (and delete[]) for C's malloc() and free().  However this is only appropriate for the allocation of simple types (byte, int, etc).  If the code also allocates objects, it's not possible to simply swap these functions out since C++ (default) initialisation is be going on.