C++ Classes – Handling Classes Used Only Once

I have a class that basically builds up a memory-efficient version of a structure to be used in a performance-critical section of code.

The resulting structure is ugly but extremely fast, so not using it isn't really an option, but I also can't use it as the only representation of my structure because it has parts not relevant to the analysis stripped. Did I mention it's also very ugly?

Let's call this class Fast. I also have the original, pretty representation, let's call it Pretty. I have a function called Process() which is supposed to take in an instance of Pretty, convert it to Fast and then do a few million calculations on it and spit out a result:

Pretty pretty;
// do stuff with pretty

// in another hpp:
double Process(const Pretty& pretty)
{
  Fast fast(pretty); // create my fast object

  // do some processing on fast

  return result; // at this point fast dies
}

The instance of Fast is created and dies within Process, but unfortunately I can't declare a class within a function, so I have to declare it elsewhere.

What I would like to do is make the entire class accessible only to Process, and not even have it seen by anyone else (just like I would conversely declare a function private if the class were the only object that should have access to it). But I can't declare a class private!

Here's my current attempt:

class Process
{
public:
  static double DoProcess(Pretty pretty); // this is the process function

private:
  class Fast
  {
    // construct fast here (only Process can see this class!)
  };
};

And then I just call Process::DoProcess(). This isn't really a critical issue so I didn't post it on SO, but it's a style question and I would really like to have a neat solution! I've never come across this issue before and was wondering how others have dealt with it, and if they came up with a nicer solution.