Is there a way to std::move std::string into std::stringstream

I do not see a std::stringstream constructor accepting rvalue reference of std::string

That's right. Even the str setter doesn't utilize move semantics, so moving a string into stringstream is not supported (not in the current standard, but hopefully in the next one).

You'll be able to move a string into a string-stream in C++20.

Move semantics are supported by the constructor:

std::string myString{ "..." };
std::stringstream myStream{ std::move(myString) };

It can also be done after construction by calling str():

std::string myString{ "..." };
std::stringstream myStream;
myStream.str(std::move(myString));