Copy std::exception object to a struct member in C++

Question

I have the following program:

#include <string>
#include <iostream>
#include <stdexcept>
#include <sstream>


enum class Severity
{
  TRACE,
  DEBUG,
  INFO,
  WARN,
  ERROR,
  FATAL,
};

struct LogRecord
{
  const Severity          mSeverity;
  const std::string       mTimestamp;
  const char* const       mFunc;
  const size_t            mLine;
  const char* const       mFile;
  std::string             mMessage;
  std::exception          mException;   


  LogRecord& operator<<(const std::exception& e)
  {
    mException = e;
    return *this;
  }

  template <typename T>
  LogRecord& operator<<(const T& msg)
  {
    std::ostringstream ss;
    ss << msg;
    mMessage += ss.str();
    return *this;
  }
};


void throw_exception()
{
  throw std::logic_error("Logic error");
}
int main()
{
  LogRecord l{};
  try
  {
    throw_exception();
  }
  catch(std::exception& e)
  {
    l << "Test" << e;
    std::cout << l.mMessage << ": " << l.mException.what() << std::endl;
  }
}

The main function prints "Test: std::exception" instead of "Test: Logic error". The reason is that "mException = e;" does not copy the exception to the mException member of my struct. What is the reason for that and how could I archieve that "Test: Logic error" is printed?