Together, we have over $200K (mostly in stock market investments), and our parents may help with the down payment.

though; (1) there are tenants in the house on a two year lease (payments are enough to cover mortgage payments), and (2) the Seller wants the option to buy back the property in two years.

The two existing LTR units (on my family’s property) were built 5–8 years ago at a cost of about $80,000 per unit, so I’m estimating a 20% increase in costs for the new builds.My plan is to use the equity in the land as the down payment for the construction loan, which I expect will make closing costs the primary upfront expense.

Find a down payment assistance program and see what that looks like. 

I use Buildium for tracking rent, fees, leases, and collecting online payments, but I don't use their reports for much, as I end up copying my online banking each year into spreadsheets.I've been using Rental Hero for the last hour and already impressed with it.

This is where the strategy of offering a higher down payment—like 25%—can make a big difference.

If you offer more $ for the property than the all cash offers AND you appear to the seller to be highly qualified in the method you are using for sourcing the payment, at some point your offer will be more appealing than the all cash offers.I don't understand why you would consider a 1031 for another property in two years. 

Even if it does, there's closing costs to consider.As I said, I have no cash myself, so I'm asking how I can deal with generating cash for closing costs (and a down payment) if it comes to it.The house costs $139000.

The emergency fund would never be part of our down payment on the next property, just the funds in the brokerage account. 

This is eerily similar to the unsecured loans I've made and haven't seen a payment on in over two years.