Disclaimer: This is coming from someone who gets a thrill out of cutting expenses, and is maybe not possible for everyone. It would take a lot of discipline to change your lifestyle to this degree.

Firstly, I think the total expenses in the op is actually $6,078 unless I punched a number in wrong.

Rent: $1,545 -> $1100 (move to an apartment or basement instead of paying rent for something bigger)
Montessori school tuition: $1,220 (no change, but I'm skeptical that this is the cheapest option)
Car payments, insurance, gas: $1000 -> $500 (sell new cars and buy two beaters. owning a single car would be even better)
Groceries and eating out: $843 -> $500 (prepare your own meals and eat out only 1-2 times a month)
Student loans: $340
Health insurance: $240
Utilities: $167
Miscellaneous: $320 (no change, but seems like a lot for someone who's struggling financially. my guess is we could easily cut $150 here)
Children's expenses (clothes, diapers, etc.): $187 -> $120 (cut down on the number of clothes you're buying and don't buy expensive brands)
Doctor copays, prescriptions, etc.: $73
Mobile: $65 -> $50 (get GoogleFi and use only 1GB per month)
Life insurance: $50
Netflix & Spotify: $28 -> $18 (cut out Spotify and listen to the ads to save $120/year)

Total: $6,078 -> $4,698

Take home pay $5,600 - $4,698 expenses = $902.

Save this amount every month and you should be able to afford a down payment on a modest home in 3-4 years. This would obviously be a big change in lifestyle, but it's not impossible. This is similar to what we did straight out of college when I was making less than $40k before taxes and living in south Provo.