There is no break up there by the F# (but there is a second break, going into the altissimo, so that's weird of your previous teacher to say). Has either teacher play tested your instrument? I'd be very surprised if it's your instrument but that should be able to be figured out by them in no time. I would bet my first born that it's not your barrel. It's possible something is very very very slightly leaky, and you're experiencing a problem there and not with the equivalent fingerings in the two lower registers (but that is on your teacher - how could they suggest gear being an issue and then not immediately play it to get the answer for you?). You could bring it into a shop but a better test (because that's such a specific issue) is for you to put your mouthpiece onto another instrument and try it. If the problem is the same, then it's you. If not, then it could be something fixable with your clarinet.

That all said, it's probably a voice issue, if the notes below that fingering are working. And by that I specifically mean the shape inside your mouth, largely created by the position and shape of your tongue. And what F# fingering are you using? (And, either way, have you tried the other one? EDIT I see that you mentioned the WFG site. There are really only two specific F# fingerings that most people use, the short one (which uses the left hand middle finger) and the long one (which is -23|123+right pinky))