brings up a couple of ideas in the O.P., which can wind up at odds with each other.
(1: The notion of saucer-separation, as applied to a Constitution-class starship
. We know that Galaxy-class starships were capable of the hulls operating independently of each other, but it was always confusing to me how a Galaxy-class saucer would be much good to its inhabitants if the saucer only offered impulse power (presumably a sublight propulsion technology) and the starship separated while in deep space. It was never made clear if a Constitution-class vessel had this capability as well. (The notion sounds plausible to me, and would make a great deal more sense than packing the hull with lifepods, but the whole notion of propulsion is still there regardless of the era.) It is never made clear in TOS, TMP or the other subsequent movies that saucer-sep would result in the stardrive section somehow being "discarded"; we really don't know what its functionality (or lack thereof) would be.
(2: The notion of how to control a starship if the main bridge is damaged/captured/destroyed
. This is also tricky territory in TOS. In TNG, it was a given that bridge control (security lockouts permitting) could be accomplished from the Battle Bridge atop the stardrive section, or from main engineering; but only while the saucer and stardrive section are united. While the Enterprise was somehow able to take control of the Starship Lantree remotely (prefix codes?), it was never made clear if/how the method could be used between a saucer and stardrive section while separated. But even more interesting, it was never made clear in TNG what would happen if a Galaxy-class saucer separated and the saucer's bridge were either damaged, captured or destroyed. How would a saucer's crew control the saucer without the bridge, particularly if the stardrive section were away on a mission?
TOS is really sketchy on all of this. Since there are strong similarities between the weapons control room ("Balance of Terror") and auxiliary control ("The Doomsday Machine", "The Changeling", "The Way to Eden"), and since the corridor sets outside of both seem to loosely suggest a saucer layout, and since ship's weapons have often been seen firing from the saucer's lower vortex, there seems to be an accumulation of circumstantial evidence that at least one auxiliary control room is positioned there with the weapons control room.
Of course, if a Connie saucer did separate, there would have to be some kind of controller rig in the engineering hull. TNG made transfer of ship's control look relatively easy; just given the command and you could transfer control to any facility in the ship with ample control panels and displays. (That's the impression I got from Riker's commands in "Power Play") Those pesky knobs, buttons, switches and proprietary displays make that TNG-style contingency problematic in TOS. (NOTE: Khan apparently seized control of the Enterprise from Engineering.)
The only thing that makes sense to me would be if there would be two control centers at opposing polar decks of the Connie saucer (main bridge is topside, auxiliary control is underside) and at least one other room in the engineering hull (engine room and/or another unseen room similar to auxiliary control; possibly called tertiary control) in case the saucer separated.
This works for me, FWIW, YMMV.