I have decided I will focus on habitat birds here, specifically ones that can be made from the three base game species, because they are what I would want most for an anniversary animal:
A new flamingo species - Could be very easily made from the greater flamingo. I would go for either the American or Chilean flamingo, to add a New World species to the game.
A rhea species - Could be made from the ostrich, but would need a number of smaller changes. The greater rhea is the more commonly-kept species globally, while the Darwin's rhea has the more unique biome (living mainly in cold mountainous habitats).
Green peafowl - Could be made from the Indian peafowl, but again would need a number of small changes to make it unique. This would add an Endangered species from a different region to the Indian peafowl.
A couple of more oddball choices derived from the peafowl:
Ocellated turkey - Have a very similar overall form and appearance to peafowl, including the fan-tailed courtship display of the male, and could quite easily be made from them. They would add a vividly coloured bird that is also an excellent addition to Latin American tropical displays.
Great bustard - These also have the fan-tailed courtship display of the peafowl, but would need an additional animation where the throat sac is enlarged. Having seen them myself in a safari park, they are hugely impressive birds, that would add a threatened species native to Europe, Central Asia and Northwest Africa.
View attachment 357801
These images show the similarity in head and beak shape between the three species (the crest of the peafowl could be repurposed into the turkey's comb, or removed altogether for the bustard).
View attachment 357802
These images show the courtship displays of the three species - all share the dropped wings and fanned tail, but the turkey would need the extending of its snood and the bustard expansion of the throat sac.
I would say all of these would be as simple or simpler to make as producing the red deer was.