What are your debts and what are the interest rates on them?
Could you do a child minding trade with another stay at home parent - they look after your kids during the work overlap and you look after their kids at some other time (so they also get a break, and it stays fair)?
Could your wife do 'under the table' babysitting work, where someone drops the kids off at your place and your wife looks after them?
Instead of ramen rice, beans or whole wheat are more nutritious, not sure if it would be cheaper though, if you get it in a 20kg bag it might be. I soak wheat then rinse it for a couple of days and then boil it as muesli for breakfast, I got my bag from a feed store - flour is also cheap.
Do you know a reliable mechanic? If you do then you could check out cheaper cars, and make sure you can check the car out with a mechanic before you buy - they can help let you know what will need fixing and how much it'll cost. Being able to get rid of the car payment and maybe even get cash back would give you a heap more leeway. I think you should reconsider getting an older car.
You could also look at station wagons instead of vans - some can have extra seats fitted in the back. I don't know if it'd be legal to put very young kids in the seats in the back but your wife could sit in it while your kids all get car seats in proper seats.
To me bankruptcy sounds sensible, given your income(any moral arguments aside), but I don't know how it will change how easy it is for you to get a better job, and it will make things harder if you're still running up debts afterwards - $30,000 doesn't sound like a huge amount of debt, except for how low your income is. If you got the electrician job it wouldn't take very long to pay off, you'd just need to keep your head above water till your pay increased.
I hope your situation improves and I'd really like to see how you pull through, good luck!