This would raise our mortgage by ~ $700/month, which is a lot, but still makes our total mortgage payment well below rental on a 4bd, 2 bath.So, my question for the great minds of BP is, is this a good idea?

Oh, one more thing to be wary of, section 8 contracts states that if you have long running issue and not fixing it, not only can they stop the rent payment for that unit, but for every unit in that property.and another one. 

This lead to piecing together payments here and there and drastically slowed the construction process and major cost overruns.

It's near impossible to cash flow when most homes are selling for above $750K, Taxes and Insurance added in the PITI payment is higher than the rents.You can buy a Duplex 2 unit for example in either FL, IN, or OH and not pay more than $200K the rents are typically $1200-$1500 a door.

Ultimately, leading to issues down the road (eviction/habitual late payments)

Some companies call this a markup for managing the payment of utility bills.

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.

My guess solely looking at your numbers is that it's an older/distressed property which means you have to account for more costs in capex. 2nd thought being it's not in the best of neighborhood then you'd have to take into consideration lower quality tenants/ neighbors and maybe increase expected vacancy and late payments.

If the business sells (it’s currently listed), the rent payments would stop, but I’d still own the property as an asset.

To qualify for a loan, focus on having a credit score of at least 620 (or 580 for FHA loans), keeping your total monthly debt under 43% of your income, and saving enough for a down payment and closing costs—about 5-10% of the home's price in total.