They will issue a separate slip when you check in with the dog. The important part is that PETC is on your ticket.

Also, regarding bags, we flew our cats in Sherpa bags, they are formally approved by a bunch of (US-based) airlines, and satisfy all the IATA requirements. They're really good quality and sturdy - we took them on several intercontinental trips, and still use them to take the animals to the vet. I'd totally recommend them. Don't worry about them not getting accepted - we flew them on LH and LX.