Changing Stem

Can I fit the Kalloy expander which says “ 22.2N” to get a bit more height?

No, you can't. That's for 1" threaded headsets/forks. Your bike has a 1 1/8" threadless headset/fork.

You could try some bars with added stack or an even steeper stem.

Is your bike too small/aggressive per chance? Maybe look for an endurance bike instead of this race bike.

Edit: Ah, it's a TT bike. That's going to be quite aggressive with drop bars on it.

No, the expander (a threaded to threadless converter) works by gripping the inside of the steerer. The stem you currently have grips the outside of the steerer but, more importantly, the setup you currently have uses the stem clamped onto the outside of the steerer to pre-load the headset bearings. The expander doesn't do that, so your headset would just be flopping around.

You can use one of these or any equivalent adaptor.

Edit, your steerer looks like it's cut pretty short, so make sure that anything you put on there has the topmost clamping bolt slightly below the top of the steerer.

The Trinity model is not designed to be ridden with drops, although there are solution to add stack it's quite possible the end result will handle funny.