I’m trying to relate the psi angle (diffracted angle) to the theta angle (angle between diffracted ray and the original non-deflected zeroth order ray.
Braggs law has two steps to it, and the first one is often omitted. The terminology "reflection" is used because the angle of the incoming beam with the planes is equal to the angle of the diffracted beam with the planes. This ensures that the path traveled from source to detector is the same, no matter which atom on the single plane scatters the beam. So the angle of deflection (between incoming and diffracted beam) is twice the angle between the beams and the plane.
This is the same for reflections in a mirror, the angle of incidence and angle of reflection are equal, and the total deflection is twice this angle.
The usual terminology is to call the angle of incidence "theta", and the angle of deflection "2 theta". For example, building a goniometer, one angle that is changed (that between incoming beam, usually fixed, and detector, which is rotated) is named "2 theta stage".
I wondered if anyone could possibly spot the fundamental mistake I’m making here?
No fundamental mistakes, just a little mistake: The total path difference is $2 q$, so for constructive interference, you need the condition $$2 q = n \lambda$$
If you plug that in, you arrive at Bragg's law. There is no need for any fancy trigonometry (like the tan(x) = sin(x) for small x), the derivation is complete once you fix that mistake. You should look up Bragg's law to make sure you know what the goal of the derivation is.