Good morning!
Congratulations on your healthy and happy chicks. Now, to keep them that way!
Most chicks sent through mail order are screened better than that, but you do get the occasional "oops". Just like in a shipment of 24 pullets, there seems to be an unfortunate cockerel who wasn't caught by the sexers. Cross beak is usually treated in a similar way by many, if they even bother.
By "treated in a similar way", I refer back to the unfortunate cockerel in a box of pullets - he's probably destined for Freezer Camp, if not just killed earlier, depending on the
chicken keeper's personal philosophies and how they deal with things like that.
I have had mild cross beak cases, and a very dramatically cross-beaked pullet. I treat my chicks the same way. I resolve to not breed that chick, then give them the early support they need to live their "best life" and go from there. I also tend to keep those random cockerels until/unless they cause unwarranted amounts of trouble. I've had situations where that "spare" cockerel ended up a flock-saver of some sort, or proved to be a Good Roo.
The mild cross beak birds I've had in the past managed quite well without help and were able to "sharpen" their beaks once they were older so that the cross beak wasn't noticeable once they were grown. They turned out to be good birds, but not in my "must save the eggs" group.
The severely cross-beaked chick I had grew up well. She looked to be almost as severe as yours. I offered her a range of food items, along with the rest of her group, and she grew up happy, healthy, and a little slower than the others. I would love to have given you a good report, but she was one of the many who disappeared while I was in the hospital two years ago. I suspect a cat (we have bobcats in addition to the ferals), raccoon, opossum, or dog got her.
While my severely cross-beaked chick did well, I have had others that did not. Even with damp food, porridge/oatmeal, cornpone/spoonbread, and other preparations (I would make a bird safer version of the standard recipes with chicken-centric ingredients), some chicks just don't figure out how to eat. Maybe there's a difference in the amount of "twist" they have to the lower beak/mandible, or maybe there are other issues, internal and external, that aren't as easily seen. It can be hard to tell.
I'm of the "let them try" school, for all my big talk, and will give the chicks as much of a chance as I can, and sorrow if they can't manage. Others will consider it a slow starvation and put the chick down. It's a philosophical decision.
If you decide to try to work with her, I suggest supportive nutrition - yes, she will probably grow slower and not as well. Offering cooked chopped egg, porridges (peas cooked/mashed into a paste, oatmeal, polenta-ish, ...) and other vitamin enriched options can help. You can make a paste out of chick
feed and a bit of
hot water, let it cool, and see how she does with it.
It looks like her tongue is firmly in her lower beak. That means she will have some trouble swallowing, but how much trouble she'll have is something you need to check. Too much trouble and she will either give up because eating is stressful and does her no good, or she'll try to eat EVERYTHING and choke. Just watch her to see what happens.
Take a little time and think about your options. What are you willing and able to do? Do you have the time and inclination to feed her with an eyedropper? Are you willing to make special chicken foods so that she can eat? If she was intended for your egg flock, or meat flock, is that still a good place for her? Are you more inclined to give her a chance, and what will you do if she can't swallow? Can you put her down and what would that mean to you?
At the very least, I suggest contacting the hatchery and letting them know one of the chicks has cross-beak. They probably won't do anything about it, considering some of the odder malformations and responses I've had over the years, but you're doing your due diligence as a customer by informing them so they can check with their policies and/or employees. It's a congenital malformation, and one that can be scary serious, but also one that can be adjusted for depending on the severity and how the chick is affected. It looks like it's a bad case, but you're right there with her and can make better informed decisions.
Best thoughts!