I don't think you can really get certain breeds but Cackle does have several "specials". It doesn't hurt to ask, and they might be able to throw a couple of your requests if available into the mix.
You sound like me in what I might do next year. You can get "Rainbow assorted pullets", or "Assorted Heavy Pullets" that could have New Hampshire, Production Red, Rhode Island Red, Cherry Egger, Buff Orpington, Black Australorp, Barred Rock, White Rock, Cinnamon Queen, Red Sex Link, Black Sex Link or Golden Commet. There is also a bantam mixed special that I might do, it depends on how well my children take care of the bantams they have now.
You sound like you might be describing the "Happy Cackle" special which includes 1 pair of geese, 1 pair of ducks, 2 turkeys, 1 rooster (at least) 10 pullets (at least) and 10 assorted. I also don't want swimming birds.
My current flock came from Cackle by way of feed store so I don't have 25. I live close enough to drive to Lebanon, MO so I might drive there next year on Spring Break and pick up my order so I don't have to have the mailing minimun.
CG