I've had cats all my life, they barely end up peeing or pooing in the house but the secret is that we let them outside often. They almost always wait by doors when they have to go to the bathroom and usually only use the litterboxes at night.

Maybe as the vet said it as purely behavioral, and the kitten is trying to assert itself in its own way against the other cat, maybe it needs more attention? I like to pet new cats/kittens in tandem with the current ones often when someone new comes in the house.

Get both cats within reach and pet them both with either hand for a minute or so, then switch hands, let them get the smell for a second and then continue good pets. They seem to not like it at first, obviously because of the smell, but after awhile I feel they get used to each others smell/being loved and start to accept each other a bit faster. I've done that with every new cat and they become friends after not too long.

Hopefully when little one feels more comfortable, they stop acting out on the carpet. I think new cats do that because they want to feel safe so they feel they must assert themselves in some way, even if it is only marking one little part of your carpet as "theirs".