First thing you will need to do is to make sure that the home is cleared of fleas, you said that your neighbor has bombed, well this usually does not work on single use basis. Clean all bedding and anywhere that the dog lays, vaccuum, and wash all of these things. Get rid of the vaccuum bag after cleaning, the fleas will remain in the bag for days and just hop out and back onto your dog. Make sure you also clean (boil) any combs, brushes etc. that you use on the dog. (fleas are like lice). Ok now I use a natural remedy on my lab (whisper) and I swear by it. Garlic....yes I said garlic. I will put in my dogs food, along with a raw egg for her coat. Fleas hate the smell of garlic, garlic seeps through the pores of the dog and fleas can't stand the smell. However this smell is not apparent to us. Nor does the dogs breath smell of garlic. Try this for a week and see if your dog is getting any relief. Also Do not bathe the dog more than once a month, you are stripping the dog of its natural oils and will cause the skin to dry out and he/she will scratch due to the dry skin. I also read that the puppy is going outside in the same area as the cat was, cats are notorius for carrying fleas, if you can take the pup to an isolated area where you have previous soaked the area with a mixture of heavy garlic and water, saturate the area and try to contain the puppy to this area while doing its business.
Hope this helps, feel free to contact me for more remedies if your dog does not get relief.