Having the core be a structural member seams pretty silly.
It's a two-way street, thanks to the action-reaction nature of structural stresses. If you sufficiently brace a KF core to keep it safe, protected, and granted the non-trivial structural integrity values of KF cores, then those braces work back against the surrounding vehicle. In other words, if there's a structural beam that exerts up to 100 tons of force to hold a KF core into place, the beam is likewise pressed back with 100 tons of force.
And when that isolated core is 50 or 95% of the ship's mass, you damn well better secure it to the rest of the structure well or sudden maneuvering will have that tough core come ripping through the rest of the ship like a battering ram.
I doubt there are spare K-F cores laying around.
In the Succession Wars era, no, there weren't a lot of spare cores lying around. But neither were there yardships. If you were going to repair one of the invaluable ships (as opposed to writing it off and building a new), you either found a spare core or built one, then jumped it to the stranded ship.
And in eras when there were yardships, it usually wasn't worth the cost of replacing core - it was about as easy to build a replacement ship.
I always though a yardship with reinforced repair facilities would remove the destroyed core along with with any other component that would interfere with jumping. Then jump away to a safe spot to wait while a spare core is constructed. Maybe even hand off the Jumpship/Warship to a space station via naval tug and let it wait for repairs there.
Read Strategic Operations pg130, "Repairing Stranded JumpShips."