If you own this as a rental then there is not homestead exemption and you'll pay much higher tax rate than a resident/owner occupant in SC. 

Is replacing the HVAC with a completely different unit the best solution?

In most cases, I don't recommend renting out your primary residence especially if you have no rental experience and that is because usually your primary residence was bought without the idea of renting it out.  

I’ve had multiple people contact me and I’m looking for possible solutions.

@Heather Bailey When house hacking with a separate rental unit, ensure proper insurance by obtaining landlord or short-term rental coverage tailored for Airbnb activities, in addition to your primary residence policy.

@Kris TohovitisI wouldn't recommend using your cash to pay off the mortgage on your primary residence since you can likely get a better return on that money by investing it.What's your buying criteria for the multi-family property?

Before throwing in the towel, exhaust every possible solution.

We are in a similar boat, however I have not yet found a solution for less than $5K even for above the ground.

I have my eye on a large single-family residence in Bethlehem, PA, but I have yet to acquire financing.

my condolences on your recent lost.there are a few ways you can do this, as mentioned above you can take a heloc on your primary residence which will give you a second mortgage, first mortgage payment wont be impacted. you can do cash out refinance which will give you one mortgage and payment. it will pay the old mortgage and give you money in your pocket to pay for the rehab.or you can do a renovation refinance which is similar to cash out refinance but we would contractor involved  as part of the refinancing and they are paid through the loan