can you get us a video? maybe upload a video to youtube then paste the link here?
When you hold his sides to your ears what do you hear? Crackling, gurgling, clunking or a whine as he struggles to breathe?
How old is your rat?
It sounds like a bad respiratory infection, he won't be able to eat anything solid because he has to stop breathing to chew on it. You could offer (if you haven';t already) soft foods he could lick up, swallow, breathe.
He will need antibiotics though, to cure this respiratory infection.