I would bet it's an upper respitory infection. Which requires antibiotics to fix.
Sure it could be heartworm but it's probably UPI, which is very common.
You need Clavamox for that, which is prescription only.
Here is what I would do;
Try here first-
http://www.animalprotectionsociety.org/links/Pet Owner Financial Aid Orgs.doc
Here is a list of places that will help people financially who need emergency vet care.
If you cannot get any help, I would then talk to local rescues, look for small private ones, you could probably find one on pet finder by searching for cats available in your area and then looking at what rescue they are from.
Talk to them and see if they will take him in and get him the proper vet care, many times they will, especially in this economy.
It will be sad for you guys and your son, but explan to him that kitty is sick and that he needs to go with these people so he can get better.
I have a close friend who had a JR terrier for about 9 years and then found out she had bladder stones. There was no way she could afford the vet, who was scaring the crap out of her telling her the dog had cancer and everything. They wanted her to spend about $3000 on the dog just in testing before they would treat her, but I was sure she only had bladder stones, which can be serious but if caught in time is usually not a huge deal.
The vet wouldn't budge. I hooked her up with a JRT rescue here and they asked her to surrender the dog in order to get treatment. She was reluctant and actually thought about having put down, actually brought her to the local shelter who agreed to put her down for free, had the dog on the table and then had a change of heart and felt selfish. It was like she was saying, "If I can't have her no one will".
She surrendered the dog to the JRT rescue. She has three kids, who were devestated but the older two understood that their doggie was sick and needed to go away in order to get better. It turned out the dog had bladder stones, which were removed and then she was placed up for adoption.
If he is sick and you have no other options, this may be your best bet.
Good luck.