You would not be auto-disqualified because you own a rental in SC.

What does your insurance provider say?

Here's an example of what I'm talking about:Purchase price: $10,000Rehab (everything new): $40,000Total "all in": $50,000 (fully financed, 15 year note, 5%)Rent: $650/month.If I apply my normal formulas to it....10% maintenance (-$65)10% management (-$65)Taxes and Insurance (-$70)5% CapEx (-$35)NOI: $415Less debt service (-$395)Profit: $20/month.We haven't factored in vacancy.These are real numbers I'm getting from some investors in my market.

The only SAFE way for you personally, is to get an umbrella insurance policy. 

I’m looking into a 6 family in NJ that is on the market for 550k...the gross annual rents are $83,400...taxes are $14,489...insurance is $2,600...utilities about $2,000...repairs and maintenance about $5,000.

Insurance covers that , but there goes 2 grand in deductibles . 

have a borrower who doesn't have insurance and is way behind on payments.

We also didn't have her get rental insurance when she moved in but what difference would that make if we had her get that?

Hence, the restrictions (valid IMO) in place (always paying mortgage insurance) will not be optimal if you make an above-average salary. 

Roofing contractors I've talked to also do not like wood shake as they state that insurance companies don't like it due to the fire hazard.