Can you clone a specific Test Suite into an existing Test Plan? Not really.
Shall you resort to Excel, getting – in this scenario – an half-baked result? No, either.
And what if you want to work with test artifacts in a script?
You should use tcm.exe instead! By design, MTM doesn’t expose all the possible capabilities, for a matter of User Experience. As it is a tool aimed mainly at testers, too many options might be confusing for the user. But that does not mean that a scenario like cloning Test Suites (not the whole Test Plan or the single Test Case) is impossible or barely possible by using tools not really fit for it - tcm.exe fills this gap.
What you’ll need to do is just calling tcm suites:
You can get the Suite ID from MTM. Even if you are cloning from/to the root of the Test Plan, it is still a Test Suite hence you have a Suite ID:
You can also override fields, and clone stuff across Team Projects. It is pretty handy.
But it isn’t over, as it has other capabilities:
I find very, very useful the possibility of running specific tests from the command-line (tcm run) and, very helpful in case of migrations and integrations, downloading the XML mapping of Bugs, resolutiontype and failuretype with tcm fieldmapping.