I have been in your shoes a few times.

I personally am generally against disclosing mental illness to an employer or using FMLA it will come with a letter from your provider and a simple google search will show them what type of Doctor they are.

Yes legally that can not fire you for wanting and/or needing accommodations. They can nit pick and find reasons to write you up that after X amount they can let a person go. Like 2 mins late punching in or going a couple mins over assigned lunch break. Missed a few days here and there having no childcare and no choice but to stay home. It’s awful but I’ve seen it time and time again. Hell my husband lost his job after being hurt at work and needing bilateral shoulder surgery. He was a liability at that point he had to change departments because the position that hurt him he wasn’t able to preform the work any longer because of life long impairments.. we did have to sue them over it. Basically I’m saying unless your union if a company wants to let a person go they will find a way.

I like BirdDancers idea of setting yourself some breaks while having to really push hard to get this work finished on time.