As said, unlikely that what you are asking is possible DIRECTLY.
Work around I would investigate...
You can kick off whatever you need to through the windows shell from TC. You should be able to set up a Python 2 environment on your machine, wrap whatever you need to do with 'arcpy' in its own Python script and call that from TC (via WshShell).
I would also see if your ArcMap Extension is accessible via .NET. It is not hard to call .NET DLLs from TC and you could do the same thing (and possibly faster than accessing it via Python) if your ArcMap extension has that interface.
