While rain rot CAN be bacterial in nature, more often it is fungal, and antibiotics therefore will do nothing.
However, I'm feeling chances are this is an allergy. Pigeon fever, from reading, doesn't quite fit, but I have seen and heard of stranger allergic reactions - a horse that presented with ERU [moon blindness], that in actual fact was severely allergic to oats [he was so allergic that his immune system started to attack his eyes; when oats were removed from his diet, his eyes went back to normal, and he lived another 15 years before being put to sleep due to an unrelated issue], is one example. Another I have heard of was allergic to something, I'm not sure what, and presented with sores on his coronets. Diarrhea is not unusual with a change in feed, even if it's just that the same kind of hay is coming from a different place. We also mustn't rule out the possibility that his feed is contaminated somehow.