BRRRR has central heat but no A/C. Should I install?

[Josh Clinard]

I am in the rehab process with my first investment property. The house is older (1917) but has what looks to be a fairly new central heating unit with full ducting. Since this is my first property and I am chosing the BRRRR method, I initially wanted to put as little into the house as possible to keep cash in my pocket for anything that may go wrong. Although the house is now 75% finished and I am thankfully well under budget. Since the house already has ducting and central heat, I began wondering just how expensive it would be to add A/C and how beneficial it would be for the refinance.

The house is 3 bed, 1 bath. Almost 1200 sqft single story. If the cost to install can be made up for, and then maybe a little on top when we refinance I may jump into it. I know I will need to call around and get quotes based on my house/area, but I was wondering if anyone could offer any input on how much adding central air may sway the refinance? Should it be worth the investment, would I be lucky to break even or should I settle for 3 or 4 window units?

I am planning to hold the house and rent it, no intentions to sell at this time. Any input is greatly appreciated!