Is it necessary to complete a full recitation (khatm) of the Holy Qur’an for oneself before completing one for the deceased, or is it permissible to dedicate it to the deceased without the reciter having first completed one for themselves?

You are not required to complete the recitation of the Qur’an for yourself before completing it for your mother. It is permissible for you to complete it and supplicate to Allah that He bestows the reward of your recitation upon your mother, and it is hoped that this will benefit her.