I'm looking for effective exercises to strengthen the entire core while minimizing stress and flexing on the lower back.
No exercises recommended here will be performed without a doctor's approval so please make recommendations freely.
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As you wrote the core, that would mean also obliques etc.
However, here are some exercises you could do:
hanging knee raises and as progression leg raises. http://www.muscleandfitness.com/workouts/abs-exercises/videos/hanging-knee-raise
turkish get ups (you can use a dumbbell instead of a kettlebell or a weight plate) http://www.bodybuilding.com/fun/turkish_get_up_training.htm
standing cable crunches as the kneeling ones are making too much pressure on lower spine.
hand stands against a wall for time instead of military presses because of potential wrong technique and causing pressure on lower back - squeeze your but and abs during the hold.
take care and do the exercises with caution.
I'm not sure what your condition is, but an exercise not very taxing on the spine for core (mostly abs) would be: cable crunch (http://www.bodybuilding.com/exercises/detail/view/name/cable-crunch)
EDIT: But you should check with a doctor for sure!
Keep up the good work!