I really like this idea. It will promote players to breed really high quality pets that will be able to compete better in the tournament.
The owners will submit their pet and match ups would be selected randomly by the server, or something like that? If allowed to choose who they battle against many people will pick the weakest pet they see to battle theirs againsts. That doesnt really show off the skills and strength of the pet when you took your Gp3 pet and picked out the Gp0 pets to battle.
I also think there should be a seperate class for each species, a best of cockatrice, best of dragon, best puma, ect instead of one top pet out the (currently) 665,008 grand total of pets. Once a month sounds about right as nine special items introduced a month doesnt sound too bad.
There is one species I think should not be allowed to participate though, the kami. For a few reasons, like how their characteristics and genes give an unfair advantage when compared to regular species. But if each species get their own class, and kami fight against kami, then this isnt too worrisome.
The other reason, the tournament should show off the progress a breeder has made with their species, how strong and powerful they have developed their line to be. They get to test out their lines in the tournament, and even be rewarded for all their effort. A kami isnt bred, it wasnt strenuously developed generation after generation by a breeder, it just wouldnt really belong in the tournament.
As for a mark on their page, if this becomes a thing, badge icons can be made to put in the pet's profiles. Much like the ribbons made for exhibitations.
*Edited to fix mistakes made while typing with phone*