Compile everything in C++'s C subset and get that working.
Start with a module, convert it in a huge class, then in an instance, and build a C interface (identical to the one you started from) out of that instance. Let the remaining C code work with that C interface.
Refactor as needed, growing the OO subsystem out of C code one module at a time, and drop parts of the C interface when they become useless.