Have you thought of riding a tandem and having the claustrophobic person serve as the stoker (the rear)? That way they can close their eyes and simply pedal, without having to worry about steering or braking.
A recumbent tandem trike would be ideal for this. Or a sociable.
I belong to an adaptive cycling club and we ride with blind individuals (amongst others) using tandems and sociables (as well as regular bicycles, etc.).
Update: With tandems ruled out, another option used by low-vision cyclists is to have the buddy, lead- (sighted) cyclist wear distinctive bright clothing, have a flag on the bicycle, or a steady red light (not flashing, which can cause seizures) on the back of their bicycle. This allows the vision-impaired cyclist to focus on the lead cyclist as they pedal behind. With people with anxiety, focusing on a non-stimulus helps shutout the other triggering stimuli, so it may help considerably.