Nick,
It would help if you described a little more what the third-party package does.
If the third-party package only deals in 2D translations then you'll need to decide how much vertical error you're willing to accept and let that drive how frequently you update the origin of the tangent plane. By the way, the tangent plane you describe would be part of the ENU (East-North-Up) or NED (North-East-Down) coordinate system... to provide a few keywords if you are in search of already-derived equations.
But if the third-party package deals in 3D translations then it might be usable in ECEF coordinates after all.
What exactly is your question? Are you asking how to convert coordinates from one local frame to a local frame at a different origin? If so then, conceptually, you will want to convert through the ECEF frame.
First, convert from local frame at origin #1 coordinates to ECEF.
Then, convert from ECEF to local frame at origin #2 coordinates.
Best,
Matthew