So this problem is a
single video? If so, identify it and where to watch it and someone here with a MBP and an

TV can try airplaying it to see if they have the stuttering problem too. If 1+ of us have the same problem, the problem is the video file and there's probably nothing you can do about it (but at least you'll know it's not your tech stuff).
Since all other videos play fine, it's probably just something in how this one video is encoded. If you can access the video file itself, run it through Handbrake with an

TV3 preset and try streaming it after it's reformatted. I'm guessing the issue is that the video if formatted such that the horsepower in other Apple tech (like the MBP) is "strong" enough to deal with it while the

TV3's processor just can't reformat it well on the fly for smooth playback.
It looks like you are saying airplaying this video from iPhone to

TV is working fine. If so, that's probably because the iPhone version of the video might be different than the version being fed to your MBP. Streaming services have become "smart" enough to feed different versions of a video based upon what kind of device is requesting the video. For example, if it's a computer, the computer might still be fed an Adobe Flash file. If it's an iDevice that can't play Flash files, it substitutes an mp4 version of the file. Note that the

TV can't play Flash either, so it needs a compatible version of the file.
However, if all of the problem is only a
single video and you really want to airplay that
one video to your

TV, perhaps save yourself a lot of trouble and just airplay that video from your phone and use your MBP for all the others that apparently work fine? Or connect your MBP to your TV for this one video and airplay other videos.
If I'm misunderstanding your post and this problem is MORE than one video file, I encourage you to share answers to the other questions asked in my prior post. You skipped over several that could be the cause. It's hard to help someone on relatively little information/feedback.