It is most likely that the dog picked this kitten up, shook him around a bit, and caused spinal trauma that has led to the the disuse of the rear limbs. If there are no outright fractures of the spine, then it is possible that a spinal disk may have herniated and is impinging on the spinal cord and blocking neuromuscular signals from the brain to reach the back legs. With no bowel movement, I would also be concerned that innervation to the colon may also be interrupted. This is not likely swimmers, as that is quite rare in cats, have not seen one verifiable case inf 12 years of practice.
It is also possible that there is no disk injury at all, and the spinal cord is just traumatized and causing a temporary neuropathy that could possibly resolve in time or with a little nudge with supportive care.
I had a case very similar to this about 3 months ago. I gave the kitten an injection of cortisone (dexamethasone) to treat a suspect trauma induced neuropathy) and some fluids under the skin (called subcutaneous fluids). The kitten had full use of its rear limbs and was able to defecate on its own again within 48 hours....we are actually getting ready to spay her soon.
As such, it may be wise to go back to your vet and suggest that treatment that worked well for my little feral patient with a similar, if not identical presentation. This of course is a different case and we canto know they are necessarily the same injury, but a cortisone injection and some fluids under the skin are both inexpensive and uninvasive...worth a try.
I applaud your effort to try to help these kittens, and wish you the very best!