Just how much biotechnology are you wanting these colonists to have?
Depending on how far the journey through space to their new home I would suggest putting any live animals, especially the birds, into cold-sleep. Birds are only productive for so long, so to waste feed and time on the trip only to have dirty, tired, paunchy, worn-out layers.
I would suggest extensive frozen egg, sperm, and embryo banks. I would definitely incubate every egg they get when they reach their destination. Not allow the hens of any breed (or species) to go broody and set as this decreases their productivity. Not sure if you could freeze bird embryos (I know you can freeze mammal embryos) but in Jurassic Park (the novel) they developed the embryo and then later put it into a synthetic egg so that that hatchlings could break out of a shell naturally.
With the mammals you would definitely need females onboard in cold sleep to receive the embryos when you reach the end of the journey. And like Dragonsdawn and The Chronicles of Pern you could plant cattle embryo into horses. On that note I remember reading in a college textbook about beef cattle that they planted a certain amount of embryos into a Simmental cow and she birthed 15 calves...at the same time.
ETA: I had to go find the book to see if there was any more info. The book is called Beef Production in the South by Stewart H. Fowler and that portion was emphasizing the importance of reproductive efficiency.
I think the frozen banks of biological material is your answer to genetic diversity. It would take up very little room and you would be able to bring the DNA to produce anything, even an elephant or blue whale if you wanted to because a live animal would be impossible do to the restraints of being on a ship.