I'd call your vet and ask. If it's split, he runs the risk of getting it caught in the carpet and possibly injuring himself or injuring the nail itself worse.
Unless you'd feel comfortable filing it where it won't cause trouble or cutting it. Usually, vets charge around $5-15 for nail trimming so I can't see a really high charge for taking care of a hangnail.
Unless you'd feel comfortable filing it where it won't cause trouble or cutting it. Usually, vets charge around $5-15 for nail trimming so I can't see a really high charge for taking care of a hangnail.