A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Wayfinding
From: Antoine Couëtte
Date: 2021 Nov 1, 11:13 -0700
To the best of my knowledge, the Doniol Formulae use only haversines, at least for the Heights . At least you can use them to solve part of your problem , i.e. the Distances, with Distance = 90° - Height.
As regards the Courses / Azimuths, I would think that any function (sine, cosine, tangent and cotangent) can be replaced by a more complex half-arc function of only haversines, which - if true - should entirely solve your query.
But since I am currently lacking my trig formulae here, I am leaving such confirmation to others.