The trick to using levers to play accidentals is to arrange the music so that you have a place where the left hand is free to flip the lever up or down.
The trick is finding the right place, where leaving your left hand free isn’t going to negatively impact the music, and where you have enough time to flip the lever before the accidental is needed, but also in a place where the string itself won’t already be ringing (or where you will have time to dampen a ringing string AND flip the lever).
Most arrangements with accidentals in them are already arranged this way, but if you do your own you need to examine the music to find the best place to put the lever flip, and arrange it so that you have the time.
With experience you will probably find that you will learn to be able to do lever flips faster and more accurately, and so you can put the lever changes closer to when the note is actually played with less time needed in the arrangement to have the left hand free.
As for actually playing them, lever changes in the music should be practiced just like every other aspect of the music. A lever change is as much a part of the music as plucking the string, or hitting a key on the piano while pressing a pedal. Don’t think of the action as separate from playing the notes. The movements and actions involved should be incorporated as part of the whole piece of the music. Your motion to move in and flip the lever should be in rhythm with the music as much as possible and flow with all your other movements in playing. It then becomes part of your muscle memory just as plucking the strings and placing your fingers ahead become part of it.
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