You need to simplify, not add complexity. If you are taking on a second job and he will still be part time, income will be a bit tighter, but time will be extremely tight. Recognizing that budgeting is not his strength, you will need to keep some of the important mental load stuff, but he can definitely do the lions share of the day to day household management. Although, if you think big spending to save is a good plan, I do question whether your financial skills are that hot, but it may be that you are in panic mode.
Whether he likes it or not, your husband needs to engage with his previous employer and maximise his chance of recouping his losses. As much as no one wants to parent a partner, if he is mentally unwell at the moment, you might have to. Write the letter for him, he signs, you send.
Look at every single regular spend - do you need it, can you reduce it? E.g. streaming services, Sky, broadband, gym membership, mobile phones, cheaper supermarket, own brands etc. Everyone has different things that are nice to have but really aren't essential. No big spending, keep the savings for emergency/urgent need, because nothing will return on the investment in time.
I never had a tumble dryer, or a heated dryer, when I had kids in nappies. All clothes were line/airer dried. Both of us had full time, high pressure jobs. My H pulled his weight on running the house though. And that is where your H can really make a difference, picking up tasks that don't need mental effort while he recovers, they are just the repeat drudgery of life tasks. Engaging more with life and his children may actually help to recover more quickly.
The main goal has to be to make this as temporary a situation as possible so he can move back to a better position and allow you to go back to one job again.