By lowering the basement floor, you would (as you have indicated) probably undermine the existing footings. What you could do is sequentially work your way around the basement, excavating small sections along the wall length so that no significant areas of footing are undermined at the same time.
For example, you could excavate a 5 foot width of wall, under the existing footings, where the current wall/footing configuration could "span" across your excavation without significant settlements. Install your deeper wall/footing in the 5 foot width, and then proceed to the next 5 foot segment. These segments can be spaced out around the perimeter of your basement so that more than one can be worked on at the same time. Just be sure that there is sufficient undisturbed footing on either side that can take up the load.
Also - always a good idea to survey your wall prior to starting the work so that you can monitor any settlements that may occur.