OP thinks the assumption of "epicycles" tips the balance of "least assumptions" to the contending theory. (He did not elaborate the full count of assumptions of either.)
I think this is a good test case for The Razzor. (tm) /g
(Not concerned here with the meta question of why even accept aphorisms as fact.)
Epicyles are observable phenomena and noted from antiquity. The Ptolemic System was an answer to the cosmic riddle that incorporated a reasonable, integral and coherent, explanation for this naturally observable phenomena. The Copernican System also explains the phenomena per its internal logic.
So Epicyle, imo, is not an assumption in either case, since it was not 'introduced' into the model. It is an inherent property of the system P and subjective side-effect (observer relative) phenomena in the system C.
The Copernican System assumes the following:
1 - Earth is not flat.
2 - Earth is rotating around its axis.
3 - Earth is rotating around the Sun (just like any other body in Solar System.)
We know these are facts today since we have proof that explains these "counter intuitive" and "unfamiliar" phenomena. We (1) experience a flat ground, (2) do not feel as if we're on surface of a moving body, and (3) clearly see :) that everything rotates around Earth!
That is the 'factual' state of affairs at some point in the Middle Ages.