template const / non const method

Suppose we have code like this:

template<class CALLBACK>
struct INIFile{
    INIFile(CALLBACK &processor) :
                    processor(processor     ){}

    bool process(){
        // lots of code here,
        // call processor
        processor(123);

        return true;
    }

    CALLBACK    &processor;
};


struct MyProcessor{
    void operator()(int val){
        // do something
    }
};

struct MyConstProcessor{
    void operator()(int val) const{ // !!!!!
        // do something
    }
};

int main(){
    MyProcessor p;
    INIFile<MyProcessor> ini(p);
    ini.process();

    // const case:
    MyConstProcessor cp;
    INIFile<MyConstProcessor> cini(cp);
    cini.process();
}

In both cases INIFile<>::process() will be non const method.

Is there an easy way to do it const in second example e.g. cp, without duplicate all logic in INIFile<>::process().