Why not buy it with private money, borrow enough for the purchase/repairs/carrying costs, and then refinance once the repairs are done?

You should also set aside money as a reserve for vacancies, turnover costs, bad renters, or capital expenditures (roof, flooring, and other big expenses).

Real estate is the long game, if you invest in 3-5 years besides the last 5-7 years you typically will not have significant equity gains, but when the govt pumps 6T into the economy that is why pricing went up and people made money, not because people became savvy investors.

I realize the cost may fluctuate as time goes by. 

That'll require you to determine how much the rehab will cost and pay that up front.

We have a decent amount of savings, at least enough for a down payment and closing costs on our next home, but the buying power just seems so limited in our area. 

There was a relatively low cost to purchase and rehad the property.

The 70% rule is a common guideline the rule of thumb is to never pay more than 70% of the after-repair value minus repair costs

Real estate is a long game, and getting a good asking price when the demand isn't as competitive because most people have similar thoughts to yours - might be worthwhile to take on the investment.

My thought in this stage of the game after I have a little bit of knowledge is, "what is the LLC really protecting?"