I am a volunteer pilot and certainly not a subject matter expert in the realm of animal rescue. However, I have flown 23 rescue flights and transported dozens of dogs to safety.
The answer to your question is dictated by law and not compassion, which is not always what we want to hear. As you know, every state has their own requirements with regard to bringing in animals over state lines. If needed you can access that information through this link: http://www.aphis.usda.gov/import_export/animals/animal_import/animal_imports_states.shtml
Personally, I will not make a transport across state lines without the proper paperwork, but I speak for myself and not for all pilot volunteers. There is too much at risk. Even at the small airports we typically fly in and out of, pilots can be "checked" by animal control officers ... and it has happened. Without the proper paperwork, health certificates etc. the pilot is at risk for being fined and the animals are at risk for being confiscated. Some states have little tolerance for non-compliance of these regulations; they are simply trying to insure illness and disease do not spread amongst the existing animal population. I would also speak to a vet about flying 4 day old puppies. I realize they are older with each passing day but that seems awfully young. You may want to check this thread: http://www.pilotsnpaws.org/forum/viewtopic.php?f=4&t=1674
Hope that is of some help and many thanks for the work you do on the ground.