Just a guess, but based on the position and angle of your seat (forwards, and angled downwards in front), your arms are taking too much weight. So you stiff arm it, which requires locking up and scrunching your shoulders which can put strain on your neck. You don't look too stretched out as your back angle is pretty neutral and the angle between your upper arm and torso looks good (close to 90 deg).

To prevent the neck problem you probably need to be able to ride comfortably with your shoulders dropped and your arms slightly bent. Core and upper body strength helps with this too, but perhaps you need a level seat positioned for more setback? And this will require bringing your bars back to meet you so you aren't too stretched out. A long reach bar like one of the Nittos being discussed might not be the best choice unless you can reduce your stem length (or TT length) by a good bit too.

What did your fitter say about the neck soreness? It sounds like you already did the seat levelness thing. How did you react to so little saddle setback? What about a second opinion on fit from a different fitter?