Hey OP, I've been there. Before my husband and I got married, he had a terrible creeping mental health crisis.
He was burnt out in a career he hated, but stuck because he was supporting his mum financially too. When he got suicidal, he got some therapy (back when NHS waiting lists were a little less horrific) and told his employer, who then (illegally) fired him. Cue a year of job hunting, rejection, one rescinded job offer because his past employer illegally disclosed his mental health issues.... we were dipping into savings for 6 months and then that was all gone. It was a horrid cycle of rejection and crushed self esteem. He took on a more domestic role in the mean time, and the house was spotless and dinner cooked. The one luxury he kept until the savings ran out was a cheap gym, so he could at least feel like he achieved something there.
I didn't know how to cope and would constantly send him jobs. I would break down and sob in my work toilets every time an interview didnt make it to an offer. We budgeted to the absolute digit. I felt like was carrying both of us, but we were slowly sinking. We had to move out of our gorgeous rented home into a small room with an en suite. I had no money, and even worse, it felt like the funny, confident, clever man I fell in love with was a shadow of his former self. I felt lonely.
Finally he got an entry level job in a career that he was desperate to break into... on half the money he had been previously. It was an ego hit, but he was elated to be working again. And we slowly started to float again.
And even though it was one of the hardest years, I am so glad we did it. He is thriving in a role he loves, making decent money. I have the person I love back, and he isn't constantly burnt out. But I will say..... he needed to fix it himself. He needed to take ownership as a man, and see us as a unit.
If he can't work full time, part time service work and taking on the brunt of childcare and household chores should get him up and about every day in a way that benefits the whole family whilst he job hunts. Your little one gets bonding time with Daddy. You get to offload some domestic load. It's not ideal, but it does doesn't have to break you both. Good luck!!