I've completed renovations and am thinking I can do a cash-out refi, take out the private lender and access some equity for another investment.

A year ago, three people leased a three bedroom apartment, and a $350 fee was charged.  

Maybe the 2br apartments are too small and will have to rent the same as 1br.  

Working with property managers, you can have access to a lot more than just 1 unit; however, they have their own system in place and as a tenant you ultimately need to respect their system.For example, you can't just throw in your own smart locks, hire a vendor to clean the carpet, or your own enhancements, etc.

I believe the break even point would be too far down the road to put in a fourth apartment.

Your DD period will allow you to verify some of that information, like cross checking the leases that you will now have access to with the rent roll provided to you prior to LOI submission.

The MLS has a lot of deals on it and you have access to some good people.

I'm looking at several apartment buildings for sale in Idaho next to a major university.

Buying apartments is a totally different ballgame vs. residential.  

I also work in different parts of town, so we're interested in something very central, hopefully in a decent (or very soon to be decent) neighborhood with good/quick access to different parts of the city.