Have you ever looked at the shape of a magnetic field from a dipole? For the first point, it is not uniquely defined. For the second point, it is highly non-uniform. Thus to calculate the force between two dipoles is extremely difficult. Further, when you have two collinear (parallel or antiparallel) dipoles, neither are interacting with the strong part of each other...they are interacting with fringe fields that are at angles that aren't perpendicular or parallel to the pole face and (second) further, the angle of the magnetic field depends on their separation.
Nothing about that lends itself to simplification.
My second thought is you seem to be equating torque and force. For instance you end with a question about the sum of the forces, which I take to mean adding the torque and the force. Since these do not even have the same units, you can't add them.
Now it is clear that there will be a term in the equations that will be substantially similar to a cosine term. However, all of the confounding factors listed in the first paragraph above will change the factor from a simple cosine term to something much trickier.
If you take a dipole and immerse it in a field that is less complicated than that associated with another dipole, you can think about doing analytic calculations. But not here.
As a bit of interesting additional information...take an introductory college text book. Look up Coulomb's law and how much problem solving is done using it. Now look for any problems with nearby magnetic dipoles and repeat the exercise. In one case, you have nearly a chapter of text and a hundred problems. In the second (magnetic case), you have zero of both. That tells you something about the tractability of the problem.