Aligning actual with target profile / rigid transform

Ok, I am playing around with this problem quite a while now. So far it was just fun and games, but now it's about time to put the game to bed. Maybe somebody from the community has encountered a similar problem or has a better mathematical understanding then I have.

My situation: I measure a geometric 2D-profile and I want to align the measured profile with a target profile (translation and rotation) to evaluate the deviation.

I tried to align the profiles with a rigid transform and the least squares method, which works well for certain profiles when every point of one profile has a corresponding point on the other profile. But in my case the target profile is just a section of the measured profile with less points. Also possible outliers might be a problem.

What is the correct approach to this problem? Do I have to leave out the rotation? I really cannot wrap my head around it. A little push in the right direction is needed.

Thanks in advance. Attached a picture (red one is the target profile)