I booked directly with UIA.
I emailed them the problem and they responded:
" Information that is necessary for special conditions of date change should contain the diagnosis, date of application to the medical institution, duration of disease.
Please, note the medical certificate must be signed by attending doctor, and stamped by doctor and hospital stamp."
Unfortunately, the letter I had already gotten from the doctor made it sound as if the problem has been ongoing for an extended time. So they denied me.
I responded that at the time I made the tickets, I didn't know he'd still be sick.
They responded that I can have a refund of the taxes of $275 per ticket.
I took their offer.