Awww, I'm sorry for your dilemma. There are a lot of factors to affect your decision, but in the end it's up to you. Each chicken-keeper has their own personal standard as to who to coddle and who to cull. Make the decision that makes the most sense to you.
You said this is the only chick who hatched, so that might convince you to cull this lonely only, or to instead hand-raise & coddle. A lot will also depend on just how severely crossed it is, how well it's getting along otherwise, how content or distressed it seems. And also on the other priorities in your life, and your purpose for keeping your chickens. You might get pleasure from caring for a "special needs chicken" or your resourses might be better spent tending to the rest of your flock.
There are other threads here on crossed-beaked birds, some who died & others who throve. Some required lots of extra care & attention, others got along all right on their own. I don't know how closely it's tied to genetics, or if it's just a random birth defect. I had one crossed-beaked bird who had many siblings with straight beaks. But if you keep it, perhaps to be safe, I wouldn't hatch its eggs nor let it mate with your hens.
There's also a thread on best methods for culling chicks, with a variety of opinions expressed there. Some folks disagreed with the method I had described, and I did some further investigation into the matter. After surveying several experienced chicken keepers at our local poultry club, I learned many more methods used for culling chicks. Some I dare not describe here, but there was one that made the most sense. PM me if you want to know more.
I wish you peace & grace & wisdom in your decision, whatever you choose...