The only way to know for sure is by checking his birth on OPR. If you have a LDS centre near you, you can view them there,
YOu can check online where your nearest would be.
On LDS site it only gives fathers name and not mothers at all.
On checking for a marriage for William King betwen 1847-1867 it gives 10 choices. only 3 of them a re likely. " you mentioned already and
14 FEB 1747 Beith, Ayr, Scotland to Janet Smith
Amanda