It is not true that everyone who makes more than £1,000 trading income will automatically be classified as a business and need a licence. You need to look at the costs incurred in raising a litter (stud fees, food, KC registration, vet fees etc.) against the income from selling the puppies. If after looking you roughly break even, then obviously you are not a business operation and so no breeding licence would be required.
However, it is important to note that a £1,000 trading income (ignoring any costs involved in producing the litter) is a tax threshold. Therefore, if you have an income of £1,000 (again ignoring any costs involved) you have an obligation to declare it to HMRC, at which point the costs incurred will also be assessed to determine whether tax is payable or not.
Defra have released the guidance document which helps clarify this.