Bulldogs are really great dogs but they certainly have more than their fair share of skin problems! I agree with you that the area you are describing does not sound like anal glands and its presence would certainly explain your dog's reluctance to defecate.
What this sounds most like is an abscess - a localized infection that creates a pus-filled pocket. This seems more likely than just a cyst since it is inflamed and seems to cause your dog some discomfort. If this is an abscess, it will need to be drained - it is also possible that it might rupture on its own, releasing the pus. If this happens, clean the area very thoroughly by flushing repeatedly with water. It is good that he is already on an antibiotic, but it is very hard for an antibiotic to penetrate into an abscess, and that is why draining itis generally necessary. Abscesses generally form as a result of a puncture wound of some sort - cat bite wounds are notorious for this, but any type of puncture wound can cause it. It is not likely that it would have developed as a result of the Staph skin infection though as these are generally involvling the superficial layers of the skin, and an abscess involved introducing bacteria into the deeper layers.
Another possibility here would be a perineal hernia, but this would occur more to one side of the other rather than right at the top of the anus. When looking directly from behind, a perineal hernia would result in a swelling from approximately the 9 - 12 o'clock position or the 12 - 3 o'clock position depending on which side was being affected. Anal glands would cause a smaller swelling at either the 4 - 5 o'clock or 7 - 8 o 'clock positions. From your description I am picturing a swelling more from may be 11 to 1 o'clock position. Let me know if what you are seeing seems different than this. A perineal hernia occurs due to the rupture of the muscles on either side of the rectum and can be quite serious if organs from inside the abdomen become entrapped in this hernia. This would be an emergency situation, so if you think this may be a perineal hernia, you should have your dog seen by a vet immediately. I do not think your description of where this swelling is located is necessarily suggests a perineal hernia, but I cannot see it to be certain of this.
Keeping the area clean is always a good idea, regardless of what is causing it. Make sure you go outside with him and if he is having trouble urinating, that would be more suggestive of tne perineal hernia as it is often the bladder that becomes trapped in this. If this is noted, have him seen immediately. It is likely that he will need to be seen by a vet anyway to have this drained - the type of fluid present will give your vet an idea of what is causing it.
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