Ok, apologies if i was a bit obscure.
"...I have not seen an airfield where a published CM is tied to a particular runway..."
Strictly speaking, this is CORRECT. If a flight is allowed to circle in a sector, the minima applies for that sector, not runway/s. Of course, if there is only one runway, this becomes academic.
A sector in the circling area where a prominent obstacle exists may be ignored for OCA/H calculations if it is outside the final approach and missed approach areas. When this option is exercised, the published procedure prohibits circling within the entire sector in which the obstacle is located.