no, the locking nut will take care of tuning stability at the headstock end. locking tuners basically just tighten the string down, so you don't have to wind it around the tuning peg multiple times. before i had a guitar with locking tuners i had always thought they helped tuning stability, but i don't think they really do any more than standard tuners
Locking tuners, for the most part, speed up string changes. They do help with tuning stability, especially with non-FR type tremolos. However since you'll be using a locking nut, the tuners are taken out of the equation as soon as the locking pads are torqued down on the nut.