Gina - Feel free to set up your business checking account with anyone. You will still be free to another bank as a lender.
Regarding lenders, it seems that smaller local banks have more lending flexibility. Try googling "local bank greensboro bank fox" for a list. I was in your shoes looking for local banks in the triangle and found that bank fox had a nice condensed list. Make a list of the banks and their phone numbers, then dial away and ask to speak with a loan officer. While I learned quite a bit using this approach, I learned that it makes more sense to have the deal first. Evidently lenders are not crazy about dealing with hypotheticals.
It's possible to purchase as an individual and use a quitclaim deed to transfer ownership to your LLC. Most banks have a due on sale clause that gives them the right to call the loan. Based on my understanding, it's rarely enforced but always possible.
Note, I'm a newbie so take this info with a grain of salt.
By the way, I'm from Greensboro and familiar with the area. Keep me in mind as a possibility if you and your partner ever want to JV on a flip. I can bring cash to the table as well.
Good luck!