Congratulations on the upcoming second child!
I'm seconding Tuskalusa's approach here. Take things one step at a time. Don't jump the gun.
It sounds like the new kid on the way must be a very recent development. So there is certainly an emotional reason to start planning out all your next steps and even slightly panicking over all the changes you'll have to make. Life happens quickly enough, but there is certainly enough time to take a breath, think things through, and choose to take each step only in its appointed season.
You need a good used car sometime before the new baby arrives in November. So go car shopping sometime in August, once a new crop of summer trade-ins hit the lots. No need to buy anything sooner than that. I agree that a used Camry is a good choice. I'm also partial to older Volvos; a used Accord would also work fine. I would stress not spending more than $10,000 on the car no matter what (which gets you a 5-7 year old Camry with less than 100,000 miles); a more ambitious thrift target of $5,000 is also quite reasonable. I generally pay 3 figures for my cars, but I'm crazy. If the current car has been fixed up and is in roadworthy shape, there is no need to panic into buying immediately just because you are expecting a second child come November.
You are contemplating moving once your older child is school aged. She will start school in 2021, I'm guessing August, so you need to start house shopping in late 2020 or early 2021-- not next month. Accordingly, in mid-2020 you will need to sit down and decide where you want to move, whether you want to rent or own in that new neighborhood, whether you want to carry through on the plan of being a landlord by retaining your existing home (which may be too much bother with two children underfoot), etc. You can start looking into these questions, but you hardly need to decide them now. And not only will your daughter's starting school cut down day care expenses, but you state that certain student loans will qualify for forgiveness by 2020, improving your balance sheet and cash flow (but do look into whether the forgiveness has any tax implications -- in many cases it will), so you will need to take that into account at the time.
One thing you can do in the immediate future is deep background research on your neighborhood options for moving. You claim that the good school districts in town all have prohibitively expensive houses while the good suburban school districts all have terrible lifestyle / commute implications. With respect to the in-town neighborhoods: do your homework carefully and also consider the buy-vs-rent decision. If you feel strongly about keeping your current job (which appears to be in town) while sending your daughter to good schools and avoiding a clownish commute, perhaps it makes sense to rent in that good neighborhood and pack away the savings until you are FIRE and can re-consider your location and school options without a job commute to worry about. Very often, relatively expensive residential neighborhoods have purchase and rental pricing trends which favor renting from a financial point of view, not purchasing.
With respect to the suburban options: It might appear that all the neighborhoods have terrible lifestyle / commute implications, at least at first glance and on average, but in many parts of America those suburban towns that are largely filled with sprawling, non-walkable subdivisions and car-centric commercial strips will have old walkable historic pockets here and there, or at least mid-century neighborhoods with bike-friendly residential streets and the ability to get to shops without driving. You don't have to live in all the suburban houses, just one. There are good odds that even in a suburban belt where most of the housing stock is annoying from a Mustachian point of view, there are still a few good houses. Is there a commuter rail station or commuter bus service that you could live close to? Could you switch jobs to one based in a suburban office park, and then strategically live close enough to bike to that?
Take your car and bike out one weekend once the weather improves a bit and start exploring, before locking your thinking into too narrow a rut. You have years to figure this part out.
Regarding your finances: I've got you down at a net worth of $450,000 (assuming the student loan forgiveness happens in 2020), which is good for your age.
But your spending habits are totally uncontrolled, or very poorly recorded. I have your take-home cash flow last year down at $85,215, which is your family's $128,000 gross salary after deducting retirement contributions ($33,000) and income tax withholding ($6,700), as well as FSA ($600) and dependent care account contributions ($5,000), which in each case I assume are applied to defray some of the spending categories you detail. I tally your recorded expenditures at $43,746, which includes mortgage (PITI), health insurance (which may actually be deducted by your employer before your paycheck is deposited but doesn't affect the math here), the portion of daycare costs above the $5,000 defrayed by the dependent care account above, and all the rest. Note that you put down health insurance at $2,400 twice -- I don't know if that was an error.
So we're looking at around $40,000 of phantom spending. I assume you're spending it all, because if you weren't you wouldn't need to be raiding your taxable brokerage to move cash to your IRA to earn your tax deduction for the year. Where's the money going? That's a mystery worthy of Agatha Christie.
So what you need to do between now and this August, when you go out and buy that used Camry or Volvo, is take a very deep dive into your spending habits and get them under control. With a gross salary now at $138,000 you should have no problem fully maxing out your tax-advantaged space this year, which appears to be $47,000 for long-term retirement savings (plus tax advantaged current spending accounts like FSA and dependent care) using new money (not money recycled from your taxable account), plus paying $13,000 in daycare charges, $15,000 toward your mortgage and $30,000 towards a semi-frugal "everything else" budget (food, clothing, car, travel, utilities, entertainment), while having plenty left over to pay your taxes, save for one-off labor and delivery costs, and put a bit back into your taxable account.
If you can commit to saving at least $50,000 a year for the next ten years you should be very well placed to FIRE while your kids are still more or less elementary school aged. But if you poke along at your current rate of about $30,000 per year you will probably be chained to your desk until your youngest graduates high school, if you are lucky.
The choice is yours.
