I would be very hesitant about this...even in healthcare, where we take care of all kinds, there is still a stigma about mental illness amongst doctors and nurses. Having a documented problem doesn't "cover your ass," it puts a target on it. I've seen this with several friends and family members, in healthcare and non-healthcare fields. You can be fired for not being able to fulfill your job duties, even if it is secondary to a medical problem, mental or physical. And to collect unemployment one must work at a facility for an extended period of time....I think 6-ish months?
OP, it sounds like your manager is really trying to work with you. What I'd recommend is to look online at the job postings for your hospital and see what other options there are for days. As others have said, it's not giving up. It's setting yourself up for success. Night shift is not okay for so many people. I know I couldn't handle it even when I worked a boring ol' factory job over a decade ago, so I don't dare attempt it now that errors could mean more than messed up embroidery on a shirt.
If you're going to commit to staying in your current job, it means committing to a solid plan that will allow you to get the rest you need. The first year of being a nurse is incredibly stressful, even without working in such a specialized and acute field. Many people don't fully understand that -- they assume that a lot of the stress is over after they pass the NCLEX and maybe that misunderstanding bleeds over to their families that assume the new nurse can take up more of the slack at home.
You have to give yourself time. If it means getting a babysitter, being over-protective of your sleep time, having your husband pull more weight for a while, not offering to help with various celebrations/holidays/events/whatever... it's all for the purpose of setting you and your family up for a better future. Make a list of things that would be helpful if they were no longer on your plate. Find a way to get rid of them, delegate them, hire someone else to help or change the schedule on which they're done for as many as possible. You'll be able to slowly add more of these back to your "list" over time....but expect you'll need help for at least a few months AFTER getting off of orientation.
Also, a trick I learned from a classmate in nursing school....to avoid some of the Mom guilt and let her young children have her attention in a manageable way, she'd get a little timer and set it for 20 minutes. During this time she was alone with the child in their room and they could do or talk about anything he/she wanted. They'd color, read books, whatever....the point was that they had 100% of Mom's attention and they loved it. She did this with each child and she reported it was really helpful. Perhaps doing something like this can help your daughter have attention in a predictable way that might help discourage the acting out to receive the attention?
Good luck with your decision.