FHA Appraisal for Refinance

I'll give some back story because I'm extremely anxious, and then get to my question below.

My husband and I have owned our home since October 2020. It was an entirely rehabbed home with updated everything. It got a good bill of health at the initial inspection. She is just over 100 years old.

We ran into an issue with my husband's vehicle and have gotten to the point of needing to pay of the thing entirely/put a lot of money into excessive repairs. (Different story for a different day.) We tried seeing what our best route for finances would be to get out of debt because of this car and pay off debt we needed to take out via Carecredit. This left credit scores in rough shape and we were advised to go for an FHA Refinancing Mortgage via our mortgage lender.

We live in Wisconsin and it's cold right now. I found an FHA Checklist online and it states FHA doesn't require A/C, but if a unit is present it must operate. Alas, upon checking ours its not working.

So my question: will a non-operational central air unit cost us our chance at this?

It's the middle of winter for us here.

I just got a promotion and a very reasonable salary increase, but it's going to take us some time to build up enough savings for a central air unit/repair -- and an operational vehicle is the first priority.

Again, I am scared shitless, so I appreciate your time. Thank you!