I can only reiterate Karin’s comments. Your education and experience should support the suggestion that an evaluation by your physician or PT would be advisable to rule out any underlying conditions that may be aggravated by continuing your routine.
The ergonomics of your driver’s seat should definitely be considered. Perhaps a lumbar support, as Karin suggested, if you’re not using one, would be worthwhile consideration. As you know, your posture while driving is also most important.
In addition to the low back/core strength exercises, I would recommend that you include a daily stretching routine that includes stretches for lower/upper back, chest & shoulders, abdominals, and hamstrings.
If a client approached me with that question, I would first ask about their exercise and health status and the history and nature of the back problem. Then ask whether she has already seen a doctor or PT about it. If yes, what was the answer. If no, why not. I would probably suggest to this client that it is not so much the driving itself but other factors that most present themselves as problems. Would a lumbar support help in the car? I would then implement a low back / core program as seem to be indicated after a musculoskeletal screening and having obtained any other information from the doctor or PT.
I do not mean to be casual about this but I think you should become your own client and follow the advice that you would give to anybody else. I see that you are an athletic trainer as well as a personal trainer and like to think that you are well acquainted with questions like this coming from your clients.
If your back is bothering you now, you will still need to address it even if you are considering to drive no more to see your clients.
I wish you good luck and good health.