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.

Right after submitting the above, I thought of something else....

You also have a lateral earth pressure condition that will push the new basement wall/underpinned wall inward.  Since you are removing earth in the basement that previously served to resist this lateral pressure, you have to consider how the new wall configuration can resist the lateral forces.

I've seen numerous old houses with basements created as per my reply above.  But what they did was put back a very large, massive buttress-type wall on the inside to help in this regard.

What you end up with is an interior basement wall with two faces - one face is the original wall face, which occurs in the upper half of the wall.  The other face is the new buttress face which occurs in the lower half of the wall and is perhaps between 18" to 36" offset from the upper face.

In addition to the above comments, you might want to consider helical piers (AB Chance Co. or Atlas Systems, Inc.) or small diameter minipiles that would be installed by a specialty contractor.  Both Atlas and Chance should have websites that will direct you to their residential underpinning products.  These could be on the order of $1000 each.