getting that many 14 awg pairs through wiremold can be done BUT I suggest you use the mid sized channel Running several speaker lines in a conduit is done all the time with no issues.
Plastic channel with a snap closed cover is the easiest to use. HOWEVER, with the weight of all that cable, the adhesive backing will NOT hold it to the wall. (I speak from experience here). I would suggest that in addition to the adhesive backing that you put a pan head 1/14" #6-8 screw with a fender washer through the back of the channel about every 8 -10 inches before putting the wires inside and securing the channel to the header above the door. Then you can put the wires in and snap it closed. Do the same for the runs down the door jamb.
BTW use Pan head screws so there are no sharp edges to gouge the speaker wire jackets. These are available at HD with Phillips head for easier installation.