The requirements for a pet offer to appear in a kingdom offer are well known in the community and have been verified multiple times by multiple different people.
For a pet offer to appear, all of the following conditions must be met simultaneously.
- The kingdom’s direct next power level must require either obtaining or leveling up pets.
- That requirement must NOT yet be completed (white, not green)
- The pet to be offered must be eligible to be offered. Cosmetic pets and faction pets are not eligible to be offered in a daily deal. (Cosmetics can be offered via flash offers for real money)
If all three of these conditions are true, then the qualifying pet becomes eligible to appear in a kingdom offer. An eligible pet is not guaranteed to appear in a kingdom offer unless there are literally no other possible kingdom offers that can be presented to a player.
While the community knows the context behind what causes something to become eligible for a kingdom offer, the community does not know (to the best of my knowledge) how the offer is picked from the pool of eligible kingdom offers for that day. Although the offer is randomly generated, different players have different quantities of eligible offers that could be rolled each day.
It is plausible that the game is assigning priorities to each type of offer and makes rolls for each type of offer in order of priority until one “succeeds” and is offered as the kingdom offer of the day. Quite possibly, the champion class xp offer may be the highest priority offer (the first roll) that the game makes, which could explain why that type of offer appears most often.