That diagnosis sounds pretty vague to me and not like a real diagnosis.. to me it sounds like "Jasper has some severe allergy that makes him have skin problems" which you already knew.

Prednisone is often prescribed for people with severe rheumatoid arthritis or other severe pain syndromes. It is also commonly used for severe allergy treatment. Most preferably it is a course that starts off at a certain dose (perhaps 20 mg), and then tapered down over a few weeks until the person is off of it. As you guessed it is a steroid. In people, long-term prednisone use can cause degrading of bones and bone mineral density, even eventual kidney or liver damage. That is long-term use however. It may at least give him some relief while you look into other options, and short-term use is generally thought to not be harmful.

I am no total expert by any means. What I know, I know from doing medical transcription. Usually doctors won't put someone on long-term prednisone therapy unless they are already quite elderly or else too miserable to enjoy quality of life otherwise.

I know all of this is continually hard on you, but the hardest of all has to be seeing him suffer with it, so I personally would probably agree to a course of prednisone to start off with to at least relieve him for now and give you some time to seek other opinions and options.