Depending on the zip code you might also check with Wells Fargo to see if they offer the down payment assistance grants in that area.Could be the buyer is just trying to save cash, so counter back with no closing costs.   

Shop around to see where you can get the coverage you want at the price you want . . . we have done well with Steadily as they shop carriers and can get what you want for the specific situation you are in. 

Lots of sensitivity to LTVs and Fico as well.

Buy in great areas that are not total gut jobs (unless you or someone you know well has experience working in construction!)

I'd also really appreciate any thoughts on choosing a big national bank vs. a smaller organization, as well as advice on how to filter through banking options to find a good fit. 

Well worth the consulting dollars.  

The area I invest in produces good cash flow and good appreciation as well.

Depending on your experience you'll likely be somewhere between 75% - 90% of the purchase price and possible up to 92.5% if you have lots of experience as well as experience in the area.

Any new loan will most probably condition for an explanation for the inquiry; my guess is this wont go over well with the new lender. 

But often times there is co-tenancy, as well.