basically what we can assume is that the cat's center of mass doesn't include the four legs, because no leg is supporting the weight of another leg. The center of mass therefore is the same now as it was before she lost a leg.
the center of mass exerts a moment upon each leg. If we can assume a general shape like so:
*'''*
| |
| |
*...* *'s are legs
and that the cat is made of a homogeneous material, the center of mass on the coronal plane (the plane on which the diagram is laid out) should be exactly in the center. The reaction forces of the legs (assumed to be of uniform length and girth) would then be exactly the same. The mass of the cat, times the acceleration due to gravity, times the radial distance from a leg, divided by the number of legs gives the moment on each leg (the calculation is somewhat simplified here because we're assuming an even distribution of weight, legs arranged in a perfect rectangle, and a CoM being in the exact center of the cat). Since the cat as a whole is not rotating, the sum of all the moments acting on one leg is zero. I won't go through the actual calculation since this is on two axes and I don't have paper and shit close by.
when one leg is missing, the reaction force previously exerted by that leg is not present anymore, so the other legs must compensate. Recall that moment force is the mass*gravity*distance. The front legs experience a small increase in load, but the rear leg must compensate for a much larger load than before, due partly to acting as a lone reaction moment for the front legs, and partly because the the missing leg no longer balances a moment.
Since the cat is not moving up or down when standing, the net vertical force must equal zero. The weight of the cat represents a downward force acting on the body, and each leg represents a reaction force pushing against the body. Any weight added must be borne by the legs.
The statics is perfectly clear in my head but without pen and paper it's hard to describe in just words and I never thought I'd end a post with this, but yeah, don't strap weights to the cat. If it was possible to rearrange the weight that was already there, it would be the best thing, aside from adding some apparatus to it to help bear the weight.