Create star map with polar projection from Right Ascension and Declination

Azimuthal equidistant direct projection, tangent to the North Pole, will let you see negative declination stars. The image is not too realistic, but it preserves distances in the direction of meridians and it is used because that property.

I don't agree with how QGIS manage non-Earth datums, so seems better to me if you compute the coordinates of the projection. But scaling the map if you have it for a 6378100 radius, to a 12 radius, is not so hard.

The projection is defined for a unity sphere, because a number may represent an angle (in radians) and the arc of that angle in the sphere. You just need to multiply for your radius (R) to get the true arc in non-unity spheres.

So: k' = c / sin(c) turns on k' = R c / sin(c). And c, the distance to the center, when the center is the pole, represents the colatitude C, where C = (π/2) - φ, being φ the latitude, that is the declination (δ in my notes) in absolute celestial coordinates for a star.

k' = R ((π/2) - δ) / sin((π/2) - δ) or k' = R ((π/2) - δ) / cos ( δ )

Then, λ_0 = 0 (longitude of the center), and φ_1 = π/2 (latitude of the center).
λ = α, the right ascension, and φ = δ, the declination.
Right ascensions go from 0 (in the spring equinox direction) to 2 π, declinations go from -π/2 to π/2.

Simplified formulas (since cos(π/2) = 0 and sin(π/2) = 1) are:

x = k' cos(δ) sin(α)
y = k' -cos(δ) cos(α)

k' isn't determined for both poles. The center of the projection is x = 0, y = 0 by definition and its antipode is not included in the domain of the transformation.