Hmm. Can you partition off the carpeted areas at least for a little while to get a routine going? This is just a theory, but I think if he couldn’t go there, he would probably pick somewhere (such as in a corner) to do the majority of his business, and then you could place his litter tray there.
Then after a while, let him back onto the carpet area and he might go play in there but still think “Oh my toilet is over there, I will go back there to poop!” Mine does an almighty dash to her loo from the carpet like a person with diarrhea or something lol.
Mine chose her own toilet area in the house, which is in a corner in the dining room, so I have put her tray there and she only goes in there even though there’s carpet in the lounge where she does all her binkies and running etc. But she decided to go on the couch one time when she realised she could jump up on it! It’s an old couch but I don’t particularly want wee all over it, so I blocked it off and she just has gone back to doing her business in the litter tray in the corner
I think they get tempted with any soft materials until they are fully toilet trained and or neutered. Mine is only a young one and not desexed yet either.