How big is the puppy? Should be small enough to fit in a carrier? Some airlines will allow a carrier that can fit under the seat to be under the seat. Alot of people with small dogs, chi's; yorkshire's, etc, utilize this option. That way your baby never touches cargo/baggage and never leaves your sight.

I think all you need to do to utilize this option is buy a ticket for the pet. Not sure of the going rate tho.