If you're worried about the carpet getting wrecked maybe get a large inexpensive rug to go over it until he's a little older?
Also I know this isn't really on topic but he's really young to be doing agility for two hours a day! A lot of trainers won't even let them do jumps until they're 12-18 months to make sure their bones and joints have had proper time to develop.
The first thing I thought of - since he seems to have lots of activity in his life - is that he smells/feels/hears something under the rug.
Do you have a crawl space? Could there be critters under the house that you don't hear? Have you had the house checked for termites? Do you have a basement? Could there be water/electric lines running under the floor/the ceiling of the basement that is causing some sort of vibration/noise?
My other thought is that he likes to dig...can you put a sandbox in the yard for him to dig? When he starts in on the carpet, take him out to his dig spot.
You say he is 7 month old and you've had him for 2 months. It is not related to teething and he has plenty of companionship, exercise, other things to chew and mental stimulation. So what's left? 1) something could be in the carpet. Unlikely because you say you've been there for years and probably have cleaned it many times, plus he is not just eating one area, he'll do it anywhere in the room and even with the bath rug. 2) Nutritional cause. He may have some nutrient deficiency, this can cause an appetite for weird things. It will not hurt to get a good dog vitamin-mineral supplement and use it regularly for a few months. If he has some deficiency it would take a bit to correct and you still would have to deal with the habit he has formed.
3) Behavioral cause. This may be the most likely. Where did he spend the first 5 months of his life? He could have acquired the preference for chewing carpet in his upbringing. Some people put old carpeting in whelping boxes, or in areas where they keep young puppies to give them good traction. One can make all kinds of guesses, but maybe you know more about his time before you got him. If you don't know, you may be able to find out.
Whatever the cause, you are still dealing with habit and, every time he is successful in eating carpet, makes the habit stronger and harder to break. When you cannot supervise him closely, crate him or confine him to non carpeted areas, while you try to figure out the rest. Interesting case.... Good luck!