c++ class template specialization, without having to reimplement everything

Yes, it's possible in C++03 with CRTP (Curiously recurring template pattern):

#include <numeric>
#include <vector>

template<typename Derived, typename T>
struct Base
{
};

template<typename Derived>
struct Base<Derived, int>
{
    int Sum() const
    {
        return std::accumulate(static_cast<Derived const*>(this)->myVector.begin(), static_cast<Derived const*>(this)->myVector.end(), int());
    }
};

template<typename T>
class A : public Base<A<T>, T>
{
    friend class Base<A<T>, T>;

protected:
    std::vector<T> myVector;

public:
    /*
    constructors + a bunch of member functions here
    */
};

int main()
{
    A<int> Foo;
    Foo.Sum();
}

The simplest and cleanest solution is to use a static_assert() in the body of a method, rejecting other types than the selected one (in the below example only integers are accepted):

#include <type_traits>  
#include <vector>

template <typename T>
class A
{
public:
    void onlyForInts(T t)
    {
        static_assert(std::is_same<T, int>::value, "Works only with ints!");
    }

protected:
    std::vector<T> myVector;
};

int main()
{
    A<int> i;
    i.onlyForInts(1); // works !

    A<float> f;
    //f.onlyForInts(3.14f); // does not compile !
}

OK CASE DEMO NOK CASE DEMO

This utilizes the fact that a compiler instantiates a member function of a class template only when one is actually used (not when the class template is instantiated itself). And with the above solution, when a compiler tries to do so, it fails due to the execution of a static_assert.

C++ Standard Reference:

§ 14.7.1 Implicit instantiation [temp.inst]

3. Unless a function template specialization has been explicitly instantiated or explicitly specialized, the function template specialization is implicitly instantiated when the specialization is referenced in a context that requires a function definition to exist. Unless a call is to a function template explicit specialization or to a member function of an explicitly specialized class template, a default argument for a function template or a member function of a class template is implicitly instantiated when the function is called in a context that requires the value of the default argument.

4. [ Example:

   template<class T> struct Z {
     void f();
     void g();
   };
   
   void h() {
     Z<int> a;     // instantiation of class Z<int> required
     Z<char>* p;   // instantiation of class Z<char> not required
     Z<double>* q; // instantiation of class Z<double> not required
     a.f();        // instantiation of Z<int>::f() required
     p->g();       // instantiation of class Z<char> required, and
                   // instantiation of Z<char>::g() required
   }
   

   Nothing in this example requires class Z<double>, Z<int>::g(), or Z<char>::f() to be implicitly instantiated. — end example ]