In your climate, I respectfully doubt that @Peter48’s suggestion would result in a well-mated queen at this time of year. You may not have any drones in your hives already, and that means that the DCA won’t have any drones for her to mate with either. I would not try to get them to make a mated queen themselves.
If the split is still strong, I would purchase a mated queen, and yes, you can introduce her at this time of year. If you have a frame of eggs and open brood, I would add that a week before introducing the queen to help suppress any laying workers. The other option would be to recombine the queenless split with a weaker hive, using the newspaper method. Better to have one strong hive going into winter, than two weak ones.
I have had excellent mated queens from www.ohbees.com and from their Big Island Queen hives too. Very gentle and productive.