With a colleague of mine I found two solutions, that solve my problem.
Solution 1 - The hacky one
The idea is to use the structure of the underlying implementation of the std::vector<double>, which consists in my case of 3 members containing 3 pointers to the data of the vector.
- start address of the data section
- end address of the data section
- address of the current maximum capacity of the data section
So I build a struct containing these three addresses and use a reinterpret_cast to a std::vector. This works with the current implementation of std::vector on my machine. This implementation can vary, depending on the installed version of the STL.
The nice thing here is, that I can use the interface of std::vector without creating it. I also do not have to copy the data into a std::vector. I could also take a just part from the initial data stored in my complex class. I can control the manipulated part, by the pointers I send to the struct.
This solves my problem, but it is a hack. I can use it, because the code is only relevant for myself. I still post it, because it could be of interest for others.
#include <iostream>
#include <memory>
#include <vector>
using namespace std;
//--------------------------------------------------------------------------//
//--- This function is given, I cannot change its signature.
bool
func_in_lib(std::vector<double>& data, double& res) {
  //--- check some properties of the vector
  if (data.size() < 10)
    return false;
  //--- do something magical with the data
  for (auto& d : data)
    d *= 2.0;
  res = 42.0;
  return true;
}
//--------------------------------------------------------------------------//
struct DataType {
  double a = 1.0;
  double b = 2.0;
  double c = 3.0;
};
//--------------------------------------------------------------------------//
ostream&
operator<<(ostream& out, const DataType& d) {
  out << d.a << " " << d.b << " " << d.c << endl;
  return out;
}
//--------------------------------------------------------------------------//
int
main(int argc, char const* argv[]) {
  int count = 20;
  //--- init and print my data
  unique_ptr<DataType[]> my_data = make_unique<DataType[]>(count);
  for (int i = 0; i < count; ++i)
    cout << my_data.get()[i];
  
  //--------------------------------------------------------------------------//
  // HERE STARTS THE UGLY HACK, THAT CAN BE ERROR-PRONE BECAUSE IT DEPENDS ON
  // THE UNDERLYING IMPLEMENTATION OF std::vector<T>
  //--------------------------------------------------------------------------//
  struct VecAccess {
    double* start = nullptr; // address to the start of the data
    double* stop0 = nullptr; // address to the end of the data
    double* stop1 = nullptr; // address to the capacity of the vector
  };
  //---
  DataType*       p_data = my_data.get();
  VecAccess       va{ &(p_data[0].a),                //points at the 'front' of the vector
                      &(p_data[count - 1].c) + 1,    //points at the 'end' of the vector
                      &(p_data[count - 1].c) + 1 };
  vector<double>* p_vec_access = reinterpret_cast<vector<double>*>(&va);
  //--------------------------------------------------------------------------//
  // HERE ENDS THE UGLY HACK.
  //--------------------------------------------------------------------------//
  //---
  double dummy = 0.0;   // this is only relevant for the code used as minimum example
  func_in_lib(*p_vec_access, dummy);
  //--- print the modified data
  for (int i = 0; i < count; ++i)
    cout << my_data.get()[i];
  return 0;
}
Update: Analyzing the assembler code of the second solution shows, that a copy of the content is performed, even though the copy-constructor of the data objects is not called. The copy process happens at machine code level.
 Solution 2 - The move semantic
For this solution I have to mark the Move-Constructor of DataType with noexcept.
The key idea is not to treat the DataType[] array as a std::vector<double>.
Instead we treat the std::vector<double> as a std::vector<DataType>. We can
then move the data into this vector (without copying), send it to the function,
and move it back afterwards.
The data is not copied but moved std::vector, which is faster. Also relevant for my case I can again take a just part from the initial data stored in my complex class.
Drawback with this solution I have to create an additional storage for the
moved data with the correct size. 
#include <iostream>
#include <memory>
#include <utility>
#include <vector>
using namespace std;
//--------------------------------------------------------------------------//
//--- This function is given, I cannot change its signature.
bool
func_in_lib(std::vector<double>& data, double& res) {
  //--- check some properties of the vector
  if (data.size() < 10)
    return false;
  //--- do something magical with the data
  for (auto& d : data)
    d *= 2.0;
  res = 42.0;
  return true;
}
//--------------------------------------------------------------------------//
class DataType {
public:
  double a = 1.0;
  double b = 2.0;
  double c = 3.0;
  // clang-format off
  DataType() = default;
  DataType(DataType const&) = default;
  DataType(DataType&&) noexcept = default;
  DataType& operator=(DataType const&) = default;
  DataType& operator=(DataType&&) noexcept  = default;
  ~DataType()  = default;
  // clang-format on
};
//--------------------------------------------------------------------------//
ostream&
operator<<(ostream& out, const DataType& d) {
  out << d.a << " " << d.b << " " << d.c << endl;
  return out;
}
//--------------------------------------------------------------------------//
int
main(int argc, char const* argv[]) {
  int count = 20;
  //--- init and print my data
  unique_ptr<DataType[]> my_data = make_unique<DataType[]>(count);
  for (int i = 0; i < count; ++i)
    cout << my_data.get()[i];
  //---
  vector<double> double_vec;
  double_vec.reserve(count * 3);
  //--- here starts the magic stuff
  auto& vec_as_datatype = *reinterpret_cast<vector<DataType>*>(&double_vec);
  auto* start_mv        = &(my_data.get()[0]);
  auto* stop_mv         = &(my_data.get()[count]) + 1;
  //--- move the content to the vec
  move(start_mv, stop_mv, back_inserter(vec_as_datatype));
  //--- call the external func in the lib
  double dummy = 0.0; // is only needed for the code of the example
  func_in_lib(double_vec, dummy);
  //--- move the content to back
  move(begin(vec_as_datatype), end(vec_as_datatype), start_mv);
  //--- print modified the data
  for (int i = 0; i < count; ++i)
    cout << my_data.get()[i];
}