I am so very very sorry
Find a good vet that knows how to treat rats and take your rats to the vet immediately.
It sounds like a secondary infection.
By the time you notice a rat is ill, you are well into a medical emergency.
Please take them to a good rat vet immediately. They need to be examined and put on medication.
If you take them back to that store, they will die.
It sounds like they need to be on baytril + clavamox or baytril + azithromycin (10 mg/Ib - I get mine from a pharmacy by vet prescription) orally for at least 6 weeks if it is helping.
Are they eating and staying well hydrated? Dehydration kills. Suggest good quality rat blocks (Oxbow or Harlan Teklad 2014), daily vegs, and while they are ill - soft foods such as cooked sweet potato, mush made by soaking some rat blocks in cool water, and organic soy infant formula thickened with baby cereal, as they are easy to eat and will help to keep them hydrated.
Can you post pictures and videos?
More information would be appreciated.
How are your remaining rats doing?
You may want to post on facebook group Real Rat Lovers Want to Know
Edit: It is important to know how to have rats humanely put to sleep because most vets do not know how and you need to make sure it is done properly, discussing it with the vet and remaining present if at all possible as that it is not done inhumanely .... rats is rendered unconscious to at least a surgical level by breathing in the gas/oxygen mix used for surgery, vet will check eye and toe reflexes in make sure the rat is deeply unconscious, when unconscious to at least a surgical level the vet will inject the rat, the rat will continue breathing in the gas/oxygen mix until dead. A conscious rat is never injected in the heart as it is so inhumane that it is actually banned in some countries.
http://www.ratshackforum.com/forum/showthread.php?t=7652
Here is some info:
http://ratguide.com/health/lower_respiratory/pneumonia.php
http://ratguide.com/health/bacteria/mycoplasma_mycoplasmosis.php
http://ratguide.com/health/viruses/sendai_virus_sv.php
http://ratguide.com/health/basics/basic_health_check.php
BTW if the first baby was contagious, then the others were already exposed and also have the same thing ..... so the vet that examined the baby knows little about rats
Please let us know what happens