I find it hard to believe that any licensed practitioner would recommend an antibiotic over the phone without ever seeing the animal.
First of all, nothing you've described clearly indicates that the animal has an infection, or if he has an infection, whether it's bacterial or viral.
I would stop trying to administer an antibiotic that has only a slim chance of helping, IF administered at full dosage and for full course, and work on getting some knowledgable eyes on the horse.
In fact, an antibiotic like Penicillin, administered partially or incorrectly, can do way more harm than good, surpressing the initial infection and setting the horse up for secondary infection.
You're near several large cities - Norman, OK, Wichita Falls, TX and Denton, TX. Keep calling vets until you've got someone who will come see the horse, or hire a trailer and take the horse to the nearest vet.