For a C++ project, create the project with CubeMX _then_ use SW4STM32 to load the project.
You can then right click on the project, in contextual menu choose “New”:”Convert to a C++ Project”.
Be careful not to use C++include files in C compilation units or it will fail - there are several posts here by myself and others on how this results in the kind of include files errors you mention. Understand how to use extern “C” to make sure you have right symbol naming for linking C++ functions into C compilation units.
The bulk of the STM32 resources will not compile as C++ although they are C++ aware and friendly (again, extern “C”) - you can use the HAL code from C++ easily but you can’t compile it with the C++ compiler.
It works, I do it every day for a living...