Hello, this is Dr. Bruce and I would like to help you with your dog.
I'm sorry to hear that he is having this issue.
From what you describe, it sounds like he definitely has a deep skin infection. It is very unlikely that he can be treated with just the topical wash that you have. However, if you would like to try to do so, it is imperative that you are able to clean out any of the pus that is is the infected areas. This would mean that you need to remove all the matted fur and rinse out the infection. Ideally, the fur should be clipped and all the mats removed entirely to allow the infected sites to dry out after rinsing.
I would strongly suggest taking him to the vet for treatment. If what the vet says about the immune system being low is true, then I would recommend having a culture of the pus taken. A culture would allow the vet to determine what specific bacteria is growing in the wound, and what specific antibiotic would be effective against that bacteria.
Your vet should prescribe a broad spectrum antibiotic for use until the culture comes back (typically 3-4 days) so that your dog can begin getting some relief immediately.
I hope this helps and that he is feeling better very soon. If you would like to continue this chat, please reply to this post and I will get back to you as soon as possible.
You can clean the wound with hydrogen peroxide once per day, then cover the wound with a triple antibiotic ointment that can be purchased at your local pharmacy with out a prescription.