Of that list by BrianBorchers, the main MINLP C++ codes that I'm familiar with are Bonmin and Couenne. Both are part of the COIN-OR project, and are open source. The remainder of the libraries in that list are closed-source, or do not have C++ interfaces. Some of the codes are in Fortran; if you can get copies, you might consider writing C++ interfaces, assuming you are willing to put in that effort.
We noted that GAMS is a good modeling system, but it requires (is it true?) us to abandon major part of our original material science code in c++.
If you really wanted to, you could write a problem formulation to a GAMS file, call GAMS, and then read the GAMS output, but doing so is clumsy. GAMS isn't really designed to do anything other than specify optimization problems, so trying to rewrite your code in GAMS isn't an option.